Algebra

Algebra Practice Test: Practice Test 24

Practice Test 24 for Algebra: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

Use polynomial division to rewrite $\frac{x^3-x^2-4x+4}{x-2}$ as $q(x)+\frac{r(x)}{x-2}$ , with $\deg(r)<\deg(x-2)$ .

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Question 1

Use polynomial division to rewrite $\frac{x^3-x^2-4x+4}{x-2}$ as $q(x)+\frac{r(x)}{x-2}$ , with $\deg(r)<\deg(x-2)$ .

$x^2+x-2+\frac{0}{x-2}$ (correct answer)
$x^2-x-2+\frac{0}{x-2}$
$x^2+x-2+\frac{4}{x-2}$
$x^2+x+2+\frac{0}{x-2}$

Explanation: This question tests your understanding of polynomial division—rewriting a rational expression (fraction of polynomials) in the form quotient + remainder/divisor, just like how 17/5 = 3 + 2/5 in arithmetic. Before dividing, check if numerator factors: let's test if x = 2 is a root of x³ - x² - 4x + 4. Substituting: 2³ - 2² - 4(2) + 4 = 8 - 4 - 8 + 4 = 0. Yes! So (x - 2) is a factor. We can use synthetic division or factor by grouping to find the other factor. Dividing (x³ - x² - 4x + 4) by (x - 2) using synthetic division with 2: Bring down 1, multiply by 2 to get 2, add to -1 to get 1. Multiply 1 by 2 to get 2, add to -4 to get -2. Multiply -2 by 2 to get -4, add to 4 to get 0. The quotient is x² + x - 2 with remainder 0. We can verify by factoring: x³ - x² - 4x + 4 = x²(x - 1) - 4(x - 1) = (x² - 4)(x - 1) = (x - 2)(x + 2)(x - 1). Wait, that gives three linear factors, not matching our division. Let me redo the division carefully: (1) x³/x = x², multiply x²(x - 2) = x³ - 2x², subtract to get x² - 4x + 4. (2) x²/x = x, multiply x(x - 2) = x² - 2x, subtract to get -2x + 4. (3) -2x/x = -2, multiply -2(x - 2) = -2x + 4, subtract to get 0. Quotient is x² + x - 2, remainder 0. Choice A correctly shows x² + x - 2 + 0/(x - 2) where the quotient is quadratic and remainder 0 indicates exact division. Choice B has a sign error in the quotient, showing x² - x - 2 instead of x² + x - 2. When we divided x² by x in step 2, we got +x, not -x, because we were dividing the positive x² term that remained after the first subtraction. Verification is your friend: (x² + x - 2)(x - 2) = x³ - 2x² + x² - 2x - 2x + 4 = x³ - x² - 4x + 4 ✓. The degree requirement (degree of remainder < degree of divisor) is what makes the division 'done': here our remainder is 0, which has no degree (or degree -∞), certainly less than the divisor's degree 1. A zero remainder means the division is exact—the divisor is a factor of the dividend!

Question 2

For the quadratic function $y=2x^2-8x+6$ , what is the $y$ -intercept (as a point) to label on the graph?

$(6,0)$
$(-6,0)$
$(0,2)$
$(0,6)$ (correct answer)

Explanation: This question tests your understanding of how to identify key features of quadratic functions and their graphs, specifically the y-intercept. The y-intercept is where the graph crosses the y-axis, which always happens when x = 0: to find it, substitute x = 0 into your function and you'll get the point (0, y-value). To find the y-intercept of y = 2x² - 8x + 6, we substitute x = 0 into the equation: y = 2(0)² - 8(0) + 6 = 6; this means the graph crosses the y-axis at the point (0, 6); it's the easiest intercept to find—just plug in 0 for x! Choice B is correct because it properly identifies the y-intercept as (0, 6) using substitution, with accurate calculation. Choice A gives just the number 2 instead of the point (0, 6); remember: intercepts are points on the graph, not just single numbers, so we need both coordinates! Quick trick for intercepts: y-intercept is always easy—just let x = 0 and calculate! And remember: intercepts are points with two coordinates, so write them as (x, y), not just single numbers.

Question 3

A town’s population starts at 1000 people and increases by 10% each year. Which exponential function models the population after $t$ years?

$P(t)=1000(0.10)^t$
$P(t)=1000(1.10)^t$ (correct answer)
$P(t)=1100(1.10)^t$
$P(t)=1000(1.00)^t$

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. Real-world clue: 'percent interest' or 'percent increase' means exponential growth with that as your r. 'Percent depreciation' or 'percent decrease' means exponential decay. The problem language often tells you what type and what rate directly—you just translate to mathematical form! The context tells us initial value is 1000 and rate is 10% increase. Converting the rate to decimal: 10% = 0.10. The growth factor is b = 1 + 0.10 = 1.10. So the exponential function is y = 1000·(1.10)^t. We can also write this as y = 1000(1 + 0.10)^t to show the rate explicitly! Choice B correctly identifies the function as y=1000(1.10)^t by showing correct reasoning. Excellent! Choice A confuses growth with decay (or vice versa): since the base 0.10 is less than 1, this is decay, not growth. An easy way to remember: bases bigger than 1 mean growing, bases between 0 and 1 mean shrinking! The form y = a(1 + r)^x makes the rate super obvious: if you see y = 500(1 + 0.08)^t, you can read the rate right off—it's 0.08 = 8%. But if you see y = 500(1.08)^t, you have to subtract 1 from the base: 1.08 - 1 = 0.08 = 8%. Same rate, just written differently!

Question 4

Two lengths are measured as 3.2 m and 8.75 m. You multiply them to find an area: $3.2\times 8.75=28.0$ (from the calculator). How should the product be reported given the measurement precision?

28 m $^2$ (correct answer)
28.0 m $^2$
28.00 m $^2$
27.99 m $^2$

Explanation: This question tests your understanding that the precision of your reported answer should match the precision of your measurements—you can't claim accuracy beyond what your measurement tools or methods allow. Significant figures tell you how precise a measurement is: 3.2 has 2 sig figs (precise to tenths), while 3.20 has 3 sig figs (precise to hundredths). When you multiply or divide measurements, the result should have the same number of sig figs as your least precise input. Example: 3.2 m × 7.856 m = 25.1392 m² → report as 25 m² (2 sig figs from 3.2). This prevents claiming the product is more accurate than the measurements that went into it! Calculating area with 3.2 m (2 sig figs) and 8.75 m (3 sig figs): 3.2 × 8.75 = 28.0 m². For multiplication/division, the result should have sig figs matching the least precise input, which is 2 sig figs from 3.2. Counting sig figs in 28.0: that's 3 sig figs, but we need only 2. Rounding to 2 sig figs: 28 m². This maintains precision consistency—we're not claiming our product is more accurate than our inputs! Choice A correctly uses 2 significant figures (28 m²) which appropriately reflects the least precise input (3.2 m with 2 sig figs). Choice B shows false precision: reporting to 28.0 m² when 3.2 m only justifies 2 sig figs in the result. While the calculator shows 28.0, that trailing zero after the decimal claims precision we don't have. Sig fig rules prevent false precision in calculations—follow them! Significant figures quick guide for calculations: Multiplication/division → result has sig figs of least precise input (3.2 × 8.467 → 27, two sig figs). These rules prevent false precision. Remember: sig figs represent measurement reliability, not arbitrary rounding! Your calculator will show many digits, but you must round to match the precision of your least precise measurement—that's the bottleneck for accuracy.

Question 5

In the expression $3(x+2)$ , what are the factors?

$3,\;x,\;2$ are the terms
$3x$ and $2$
$3$ and $(x+2)$ (correct answer)
$x$ and $2$ are the factors

Explanation: This question tests your understanding of the parts of algebraic expressions—specifically, how to identify factors. Factors are parts of an expression that are multiplied together: when you see multiplication (either with · or parentheses next to each other), you're looking at factors. For example, in 4(x + 3), the factors are 4 and (x + 3) because they're being multiplied. In 3(x + 2), we can see what's being multiplied together: it's 3 and (x + 2). This means the factors are 3 and (x + 2). Remember, factors are connected by multiplication, while terms are connected by addition and subtraction. Choice A is correct because it properly identifies the factors as 3 and (x + 2), following the definition that factors are the parts being multiplied. You've got it! Choice B confuses terms with factors: it lists 3x and 2 which would be terms if expanded, but the question asks for factors in the given form. Remember: terms are added/subtracted, factors are multiplied! Think of terms as 'ingredients being added together' and factors as 'ingredients being multiplied together.' In 2(x + 3), you're multiplying 2 times (x + 3), so those are factors. But when you expand to 2x + 6, you're adding 2x and 6, so those are terms.

Question 6

The function $f(x)=3(x+1)(x-5)$ is in factored form. What does this form reveal about the graph’s zeros and axis of symmetry?

Zeros: $x=-1, x=5$ ; Axis of symmetry: $x=2$ (correct answer)
Zeros: $x=1, x=-5$ ; Axis of symmetry: $x=-2$
Zeros: $x=-1, x=5$ ; Axis of symmetry: $x=-2$
Zeros: $x=-1, x=5$ ; Axis of symmetry: $x=4$

Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Factoring a quadratic into the form f(x) = a(x - p)(x - q) immediately reveals the zeros (x-intercepts) at x = p and x = q: these are where the parabola crosses the x-axis. From the zeros, you can also find the axis of symmetry—it's the vertical line exactly halfway between the zeros at x = (p + q)/2, and the vertex sits on this axis! The function f(x) = 3(x + 1)(x - 5) is already in factored form. Setting each factor to zero: x + 1 = 0 gives x = -1, and x - 5 = 0 gives x = 5. These are our zeros! The axis of symmetry is at x = (-1 + 5)/2 = 4/2 = 2, exactly halfway between the zeros. Choice A correctly reads the factored form showing zeros at x = -1, 5 and axis at x = 2. Choice B has a sign error in reading the zeros: from (x + 1)(x - 5), the zeros are x = -1 and x = 5 (not x = 1 and x = -5). Remember: (x - p) = 0 gives x = p, and (x + 1) = 0 gives x = -1, so watch the signs! It's easy to mix this up. The sign trick for factored form: if you have (x - 5), the zero is x = 5 (same sign); if you have (x + 1) = (x - (-1)), the zero is x = -1 (opposite sign). The zero always has the opposite sign from what appears in the factor. This trips everyone up at first—practice makes it automatic!

Question 7

Solve and check for extraneous solutions:

$\frac{3}{x-2}=2$

$x=1$
No solution (extraneous)
$x=\frac{7}{2}$ (correct answer)
$x=\frac{1}{2}$

Explanation: This question tests your ability to solve rational equations and identify extraneous solutions—solutions that emerge from the solving process but don't actually satisfy the original equation. When solving rational equations (equations with variables in denominators), we multiply both sides by the LCD to clear all the fractions, which gives us a polynomial equation to solve. However, this multiplication can introduce extraneous solutions: if the LCD contains a factor like (x - 2) and our solution is x = 2, it's extraneous because it makes the original denominators zero (undefined!). Always check that solutions don't make any denominator in the original equation equal zero. Solving $\frac{3}{x-2}=2$ : (1) Multiply both sides by (x-2): $3 = 2(x-2)$ . (2) Distribute: $3 = 2x - 4$ . (3) Solve: $7 = 2x$ , so $x = \frac{7}{2}$ . (4) Check in original: Does $x = \frac{7}{2}$ make the denominator zero? $\frac{7}{2} - 2 = \frac{3}{2} \neq 0$ . Good! Verify it satisfies equation: $\frac{3}{\frac{3}{2}} = \frac{3 \cdot 2}{3} = 2$ . ✓ Final answer: $x = \frac{7}{2}$ . Choice C correctly solves to get $x = \frac{7}{2}$ and verifies it doesn't make the denominator zero, confirming it's a valid solution. Choice A gives $x = 1$ , which would make the left side $\frac{3}{1-2} = \frac{3}{-1} = -3$ , not 2. This is an arithmetic error—always double-check your algebra! The rational equation solving recipe: (1) Find the LCD of all denominators, (2) Multiply EVERY term (both sides, all terms) by the LCD—fractions will cancel, (3) Solve the resulting polynomial equation, (4) Check each solution: does it make any original denominator equal zero? If yes, it's extraneous—reject it! If no, verify it satisfies the original equation. Keep only valid solutions. The checking step is non-negotiable!

Question 8

Solve using elimination: $\begin{cases} 2x+3y=12 \\ 2x-y=4 \end{cases}$ What is the solution $(x,y)$ ?

$(3,2)$ (correct answer)
$(1,4)$
$(2,3)$
$(3,-2)$

Explanation: This question tests your ability to solve systems of linear equations—finding the (x, y) pair that makes both equations true at the same time. The elimination method (also called addition method) works by adding or subtracting the equations to make one variable disappear: if you have 2x + 3y = 12 and 2x - y = 4, subtracting the second from the first gives 4y = 8 because the 2x terms cancel out. Then solve for y, and use that to find x! Let's subtract the second equation from the first: (2x + 3y) - (2x - y) = 12 - 4, which gives us 2x + 3y - 2x + y = 8, simplifying to 4y = 8, so y = 2. Now substitute y = 2 into the second equation: 2x - 2 = 4, so 2x = 6, and x = 3. Therefore, the solution is (3, 2). Choice A is correct because when we check (3, 2) in both equations, we get 2(3) + 3(2) = 6 + 6 = 12 ✓ and 2(3) - 2 = 6 - 2 = 4 ✓, confirming both equations are satisfied. If you got (2, 3), you might have mixed up which value was x and which was y—remember to keep track of your variables carefully! Always check your answer by plugging both x and y into BOTH original equations. If you get true statements (like 12 = 12 and 4 = 4), you're correct! If even one equation doesn't work, there's an error somewhere. This check habit catches almost all mistakes and builds confidence!

Question 9

The sequence $a_n$ is defined recursively for $n=1,2,3,\dots$ by $a_1=3$ and $a_{n+1}=3a_n$ . What is the next term after $3, 9, 27$ ?

$54$
$9$
$81$ (correct answer)
$30$

Explanation: This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A recursive definition tells you how to get each term from the previous term(s): it gives you a starting value (like a₁ = 3) and a rule for finding the next one (like aₙ₊₁ = 3aₙ, which means 'multiply the previous term by 3'). To find any term, you start at the beginning and apply the rule step by step. The sequence starts with 3, then 9 (33), then 27 (93), so the next term is 273 = 81. Building sequences recursively is like climbing stairs—each step depends on the one before! Choice C is correct because it accurately calculates to get 81 by correctly evaluating each term following the recursive rule. Great work following the pattern! Choice B gets the first few terms right but makes an arithmetic error when calculating the fourth term, computing something like 272=54 instead of 27*3=81. With recursive sequences, one arithmetic slip affects all the terms after it! When working with recursively defined sequences, always start by writing down the initial term(s) clearly, then apply the rule one step at a time, calculating each new term before moving to the next. It's like following a recipe—don't skip steps!

Question 10

Given $g(x) = x^2 + 2x - 8$ , what is $g(0)$ ?

$-6$
$8$
$0$
$-8$ (correct answer)

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. To evaluate a function like f(x) = x² + 2x - 8 at a specific value, we replace every x with that value and calculate: f(0) means substitute 0 for x, giving (0)² + 2(0) - 8 = 0 + 0 - 8 = -8. Starting with g(x) = x² + 2x - 8 and finding g(0), we substitute 0 for x everywhere: g(0) = (0)² + 2(0) - 8. Now we calculate step by step: 0 + 0 - 8 = -8. Choice A is correct because it properly substitutes 0 for x in the function and calculates accurately: 0 + 0 - 8 = -8. Nice work if you got this! Choice B is a common slip-up: it substitutes correctly but then makes an arithmetic error, calculating 0 + 0 - 8 as +8 instead of -8. Double-checking your arithmetic is always a good idea! Think of a function as a machine: you put in an input (x = 0), the machine follows its rule (x² + 2x - 8), and out comes an output (-8). The notation f(0) just means 'what does the machine output when I feed it 0?'

Question 11

Consider the rational equation $\frac{x+4}{x^2-4} - \frac{2}{x+2} = \frac{1}{x-2}$ . Before solving, a student should recognize that this equation is undefined when:

$x = 2$ only, since this makes the denominator $x-2$ equal to zero
$x = -2$ only, since this makes the denominator $x+2$ equal to zero
$x = 2$ or $x = -2$ , since these values make various denominators equal to zero (correct answer)
$x = 4$ or $x = -4$ , since these values make the first fraction's denominator zero

Explanation: We need to identify all values that make any denominator zero. The first term has denominator $x^2 - 4 = (x-2)(x+2)$ , which equals zero when $x = 2$ or $x = -2$ . The second term has denominator $x + 2$ , which equals zero when $x = -2$ . The third term has denominator $x - 2$ , which equals zero when $x = 2$ . Therefore, the equation is undefined when $x = 2$ or $x = -2$ . Choice A misses the restriction from $x^2 - 4$ and the term with $x + 2$ . Choice B misses the restriction from $x^2 - 4$ and the term with $x - 2$ . Choice D incorrectly factors $x^2 - 4$ .

Question 12

Add and simplify (the sum of two rational expressions is a rational expression): $\frac{3}{x}+\frac{2}{x+1}$

$\frac{5}{x(x+1)}$
$\frac{5x+3}{x(x+1)}$ (correct answer)
$\frac{5x+2}{x(x+1)}$
$\frac{3}{x+1}+\frac{2}{x}$

Explanation: This question tests your understanding of how to add rational expressions—algebraic fractions that work just like regular fractions but with variables. Adding or subtracting rational expressions requires a common denominator, just like with 1/2 + 1/3: we find the LCD (least common denominator), rewrite each fraction with that denominator, then add or subtract the numerators while keeping the denominator the same. The tricky part is finding the LCD when denominators have variables! For 3/x + 2/(x+1), the LCD is x(x+1) since these factors share no common factors. We rewrite: 3/x = 3(x+1)/[x(x+1)] = (3x+3)/[x(x+1)] and 2/(x+1) = 2x/[x(x+1)]. Now we can add: (3x+3)/[x(x+1)] + 2x/[x(x+1)] = (3x+3+2x)/[x(x+1)] = (5x+3)/[x(x+1)]. Choice B correctly adds the fractions and simplifies to (5x+3)/[x(x+1)] by finding the LCD and combining numerators. Great work! Choice A incorrectly adds to get just 5 in the numerator, C has an arithmetic error getting 5x+2, and D shows the original unsimplified expression. For adding/subtracting: (1) Factor all denominators to see what you're working with, (2) Find LCD by taking each factor to its highest power, (3) Multiply numerator and denominator of each fraction by what's needed to get LCD, (4) Add/subtract numerators, (5) Simplify if possible. It's exactly like 1/6 + 1/4: LCD = 12, rewrite as 2/12 + 3/12 = 5/12, just with variables!

Question 13

Compare $f(x)=100x$ (linear) and $h(x)=1.1^x$ (exponential). Which statement is true about their long-term behavior?

For sufficiently large $x$ , $h(x)$ exceeds $f(x)$ . (correct answer)
For sufficiently large $x$ , $f(x)$ exceeds $h(x)$ .
$f(x)$ and $h(x)$ grow at the same rate, so neither exceeds the other for large $x$ .
$h(x)$ never exceeds $f(x)$ because $1.1$ is close to $1$ .

Explanation: This question tests your understanding of a fundamental principle in mathematics: exponential functions eventually grow faster than any polynomial function—even very high-degree polynomials—when we look at large enough x-values. The growth hierarchy for large x-values is: exponential > polynomial > linear. Even a slow exponential like (1.01)^x will eventually exceed a fast polynomial like x^100 if we go far enough to the right. This happens because exponential growth is multiplicative (multiply by the same factor repeatedly), which compounds much faster than polynomial growth, which is additive-based (even if accelerating). Comparing f(x) = 100x (linear with a large coefficient) and h(x) = 1.1^x (exponential with a small base): At x = 10, f(10) = 1000 while h(10) = 1.1^10 ≈ 2.59, so the linear is much larger initially. At x = 50, f(50) = 5000 while h(50) = 1.1^50 ≈ 117.4, still linear dominating. But at x = 100, f(100) = 10,000 while h(100) = 1.1^100 ≈ 13,781, the exponential has taken over! By x = 200, f(200) = 20,000 while h(200) = 1.1^200 ≈ 189,905,276—the exponential dominates completely! Choice A correctly states that for sufficiently large x, h(x) exceeds f(x), recognizing that even a slow exponential like 1.1^x will eventually overtake any linear function, no matter how large the linear's coefficient. Choice D claims h(x) never exceeds f(x) because 1.1 is close to 1. This misunderstands exponential growth! As long as the base is greater than 1, the exponential will eventually dominate any polynomial (including linear). The fact that 1.1 is close to 1 just means it takes longer to overtake, but it WILL overtake! Real-world insight: this is why compound interest (exponential) is so powerful long-term compared to simple interest (linear), and why viral spread (exponential) is so concerning compared to linear spread. The eventual dominance of exponential growth has huge implications in finance, biology, technology, and many fields. Understanding this mathematically helps you understand the world!

Question 14

A theater is choosing how many balcony seats and floor seats to sell. Let $x$ = balcony tickets and $y$ = floor tickets. The theater can sell at most 300 total tickets: $x+y\le 300$ . Fire code requires at least 120 floor tickets to be sold for staffing plans: $y\ge 120$ . Also, $x\ge 0$ and $y\ge 0$ . Which point is nonviable?

$(150,120)$
$(100,200)$
$(210,110)$ (correct answer)
$(0,120)$

Explanation: This question tests your ability to translate real-world constraints into mathematical inequalities (and equations) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in the context. A solution is viable (feasible) if it satisfies EVERY SINGLE constraint AND makes sense in the real-world context (like no negative quantities, whole items when you can't buy half an item, etc.). If even one constraint is violated, or if the solution is unrealistic, it's nonviable. Think of constraints like security checkpoints—you need to pass through all of them! Let's check each point against all constraints: For (150, 120): Total tickets: 150 + 120 = 270 ≤ 300 ✓, Floor minimum: 120 ≥ 120 ✓, Non-negative: both ≥ 0 ✓ → viable. For (100, 200): Total tickets: 100 + 200 = 300 ≤ 300 ✓, Floor minimum: 200 ≥ 120 ✓, Non-negative: both ≥ 0 ✓ → viable. For (210, 110): Total tickets: 210 + 110 = 320 ≤ 300 ✗ (320 > 300), Floor minimum: 110 ≥ 120 ✗ (110 < 120) → nonviable (violates both!). For (0, 120): Total tickets: 0 + 120 = 120 ≤ 300 ✓, Floor minimum: 120 ≥ 120 ✓, Non-negative: both ≥ 0 ✓ → viable. Notice how (210, 110) fails because it violates BOTH constraints—too many total tickets AND not enough floor tickets! Choice C is correct because it correctly identifies the point (210, 110) as nonviable due to violating the total tickets constraint (320 > 300) and the floor tickets minimum (110 < 120). Choice A would be viable but (150, 120) actually satisfies all constraints: 150 + 120 = 270 ≤ 300 and 120 ≥ 120. Don't be fooled—a point needs to violate at least one constraint to be nonviable! The viability-checking procedure: Make a checklist of every constraint. For each one, substitute the point and check if it's satisfied. Write 'Yes' or 'No' next to each constraint. If even one 'No' appears, the solution is nonviable—identify which constraint(s) failed. When checking viability, substitute carefully: if the point is (210, 110), that means x = 210 and y = 110. Substitute those values into EVERY inequality and equation. It's tedious but necessary—one missed check could mean accepting an infeasible solution!

Question 15

Let $f(x)=x^2-1$ and $g(x)=x+3$ . Compute $f(g(0))$ .

$8$ (correct answer)
$2$
$-1$
$3$

Explanation: This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. To evaluate a composition at a specific number like $f(g(0))$ : Step 1 - find $g(0)$ by substituting 0 into g; Step 2 - take that answer and substitute it into f. If $g(0)$ gives some value, then you need $f$ (that value). It's like a relay race where g passes its output to f! To find $f(g(0))$ , we work inside-out: First, $g(0) = 0 + 3 = 3$ . Now we take this result and plug it into f: $f(3) = 3^2 - 1 = 9 - 1 = 8$ . So $f(g(0)) = 8$ . Two separate evaluations, one after the other! Choice A correctly evaluates the composition by working inside-out, giving 8. Choice B might result from only evaluating the inner function $g(0) = 3$ but forgetting the second step of plugging that into f. You're not done until you've applied both functions in order! For evaluating at a specific number: do it in TWO separate steps: Step 1: Find $g(\text{input}) = 3$ . Step 2: Find $f(3) = 8$ . Write down the intermediate answer from Step 1 before moving to Step 2—this prevents mistakes and makes checking easier!

Question 16

A ball’s height (in feet) after $t$ seconds is $h(t)=-t^2+10t+4$ . Use completing the square to find the maximum height and when it occurs.

Maximum height $29$ at $t=5$ (correct answer)
Maximum height $25$ at $t=5$
Minimum height $29$ at $t=5$
Maximum height $29$ at $t=-5$

Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Completing the square transforms a quadratic into vertex form f(x) = a(x - h)² + k, which reveals the vertex at (h, k) instantly—no calculation needed once you're in this form! The k-value is the maximum (if a < 0, opens down) or minimum (if a > 0, opens up), and the axis of symmetry is the vertical line x = h through the vertex. In this context where h(t) models the ball's height over time, completing the square reveals the maximum height of 29 feet occurs at t = 5 seconds. The vertex form tells us the extreme value—crucial for understanding the real-world situation! To complete the square: h(t) = -(t² - 10t) + 4 = - (t² - 10t + 25 - 25) + 4 = - ((t - 5)² - 25) + 4 = - (t - 5)² + 25 + 4 = - (t - 5)² + 29. Choice A correctly completes the square to get - (t - 5)² + 29 showing maximum height 29 at t=5. Choice B makes an error completing the square: it calculates (b/2)² as 25 but forgets to add back the +4 properly, getting 25 instead of 29—after -(-25) it's +25 +4=29! For applied problems: zeros often mean 'when does quantity reach zero' (ball hits ground, profit = 0, etc.), and vertex often means 'what's the best/worst outcome' (maximum height, minimum cost, etc.). Translate the math features (zeros, vertex) into context language (when, how much, what's optimal) to fully answer the question!

Question 17

The sequence $2,\ 5,\ 8,\ 11,\ \dots$ is a function with domain $n=1,2,3,\dots$ . How is this sequence defined recursively?

$a_1=2,\ a_{n+1}=a_n+3$ (correct answer)
$a_1=5,\ a_{n+1}=a_n+3$
$a_1=2,\ a_{n+1}=a_n\cdot 3$
$a_1=2,\ a_{n+1}=a_n-3$

Explanation: This question tests your understanding that sequences are special functions with integer domains, and how to work with recursively defined sequences. A sequence is just a function where the inputs are whole numbers (like 1, 2, 3, ...) and each output is called a term: the first term a₁, the second term a₂, and so on. We can think of a sequence as a list of numbers where each number's position (1st, 2nd, 3rd) is its input! Looking at the sequence 2, 5, 8, 11, ..., let's find the pattern between consecutive terms: from 2 to 5, we add 3. From 5 to 8, we add 3 again. From 8 to 11, we add 3 once more. This pattern holds for all consecutive terms, so the recursive rule is aₙ₊₁ = aₙ + 3, with starting value a₁ = 2. Choice A is correct because it correctly identifies the pattern as adding 3 each time and starts with the right initial value of 2. Great work following the pattern! Choice B has the wrong starting value, using a₁ = 5 instead of a₁ = 2. The starting value is crucial in recursive sequences—like the first domino that sets everything else in motion! To write a recursive rule from a sequence, compare consecutive terms: What's happening from term to term? Are we adding the same number (arithmetic)? Multiplying by the same number (geometric)? Once you spot the pattern, write it as aₙ₊₁ = [rule using aₙ], and don't forget to state your starting value!

Question 18

Rewrite the expression $(2^3)^t$ using the power-of-a-power property $(b^a)^c=b^{ac}$ .

$2^{3t}$ (correct answer)
$2^{t+3}$
$6^t$
$8^{t^3}$

Explanation: This question tests your ability to use exponent properties to transform exponential expressions into equivalent forms that reveal information like interest rates at different time scales (annual, monthly, quarterly, etc.). The power-of-a-power property says (b^a)^c = b^(ac): when you raise a power to another power, you multiply the exponents. This fundamental property follows from what exponents mean: (2^3)^t means "take 2^3 and raise it to the t power," which is the same as multiplying 2^3 by itself t times, giving us 2^(3t). To simplify (2^3)^t using the power-of-a-power property: we have a power (2^3) being raised to another power (t). According to (b^a)^c = b^(ac), this equals 2^(3·t) = 2^(3t). We multiply the exponents 3 and t to get 3t. This makes sense: 2^3 = 8, so (2^3)^t = 8^t, and since 8 = 2^3, we have 8^t = (2^3)^t = 2^(3t). Choice A correctly applies the power-of-a-power property (b^a)^c = b^(ac) to get 2^(3t). Choice B uses the wrong exponent property: it applies (b^a)^c = b^(a+c), writing 2^(t+3), but the correct property is (b^a)^c = b^(ac)—you multiply the exponents, not add them! This is a very common mix-up with exponent rules. The exponent properties work because exponents represent repeated multiplication: b^3 means b·b·b. So (b^3)^2 = (b·b·b)·(b·b·b) = b^6, which matches b^(3·2) from the power-of-a-power rule. These properties aren't arbitrary—they follow from what exponents fundamentally mean! Equivalent expression check: after transforming, verify equivalence by testing a value. If you transformed (2^3)^t to 2^(3t), try t = 2: (2^3)^2 = 8^2 = 64 and 2^(3·2) = 2^6 = 64. Match! This confirms your transformation is correct.

Question 19

A population is modeled by $P(t)=12{,}000\cdot(0.98)^t$ , where $t$ is in years. Which statement is correct?

Exponential growth at 2% per year
Neither; the base $0.98$ means 98% decay per year
Linear decay at 0.98 people per year
Exponential decay at 2% per year (correct answer)

Explanation: This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent decay means the quantity is multiplied by the same factor between 0 and 1 each time period: if a car depreciates by 15% per year, it's multiplied by 0.85 each year (since keeping 85% = 1 - 0.15 = 0.85 means 'you lose 15%'). The quantity shrinks exponentially, approaching but never quite reaching zero. Looking at the function P(t)=12,000·(0.98)^t: the base 0.98 is less than 1, indicating exponential decay. The percent rate is calculated from r = 0.98 - 1 = -0.02 = 2% decay. This means each time t increases by 1, P is multiplied by 0.98, which is a 2% decrease. Choice B correctly identifies this as exponential decay at 2% per year because the base <1 and |r|=0.02 confirms the rate. Choice D has the percent rate wrong: the base 0.98 doesn't mean 98% decay. When b = 0.98, we subtract 1 to get the rate: 0.98 - 1 = -0.02 = 2% decay. The base includes the remaining 98% (the '0.98'), so the decay is 2%, not 98%! To find percent rate from a growth/decay factor: (1) Identify b (the base or factor), (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: b = 1.12 → r = 0.12 → 12% growth. For decay: b = 0.95 → r = -0.05 → 5% decay (we usually state as positive '5% decay' rather than 'negative 5%'). The subtraction of 1 is the crucial step!

Question 20

In the expression $-8n^2+3n+10$ , what is the coefficient of $n^2$ ?

$8$
$-8$ (correct answer)
$-8n^2$
$3$

Explanation: This question tests your understanding of the parts of algebraic expressions—specifically, how to identify coefficients. A coefficient is the numerical part of a term that's multiplied by the variable(s): in the term 3x², the coefficient is 3 because it's the number being multiplied by x². Remember that coefficients include their sign, so in -5x, the coefficient is -5, not 5. To find the coefficient of n² in -8n² + 3n + 10, we look for the term that contains n². That term is -8n². The coefficient is the number part that's multiplied by the variable, which is -8 with the sign. Choice B is correct because it properly identifies the coefficient as -8, following the definition that coefficients include their sign. You've got it! Choice A is close, but it forgets to include the sign: the coefficient is -8, not 8. The minus sign is part of the coefficient! This is a super common mistake, so watch out for it. To find a coefficient, first locate the term with the variable you're looking for, then identify just the number part (including the sign). If you don't see a number, the coefficient is 1 (for terms like n) or -1 (for terms like -n). Write out that 'invisible 1' when learning, and it'll help!

Question 21

Which of the following is equivalent to $\sqrt{x^3} \cdot \sqrt[4]{x}$ when $x \geq 0$ ?

$x^{\frac{5}{4}}$
$x^{\frac{7}{4}}$ (correct answer)
$x^{\frac{3}{2}}$
$x^{\frac{13}{4}}$

Explanation: When you see radical expressions multiplied together, you're working with exponent rules in disguise. The key insight is converting radicals to fractional exponents, then using the rule that when multiplying powers with the same base, you add the exponents. First, convert each radical to exponential form. Remember that $\sqrt{x^3} = (x^3)^{\frac{1}{2}}$ and $\sqrt[4]{x} = x^{\frac{1}{4}}$ . Using the power rule $(a^m)^n = a^{mn}$ , we get $\sqrt{x^3} = x^{3 \cdot \frac{1}{2}} = x^{\frac{3}{2}}$ . Now multiply: $x^{\frac{3}{2}} \cdot x^{\frac{1}{4}}$ . When multiplying powers with the same base, add exponents: $\frac{3}{2} + \frac{1}{4}$ . To add these fractions, find a common denominator: $\frac{3}{2} = \frac{6}{4}$ , so $\frac{6}{4} + \frac{1}{4} = \frac{7}{4}$ . Therefore, the answer is $x^{\frac{7}{4}}$ , which is choice B. Looking at the wrong answers: Choice A ( $x^{\frac{5}{4}}$ ) results from incorrectly adding $1 + \frac{1}{4}$ instead of $\frac{3}{2} + \frac{1}{4}$ . Choice C ( $x^{\frac{3}{2}}$ ) is just the first term, suggesting you forgot to include the second radical entirely. Choice D ( $x^{\frac{13}{4}}$ ) might come from multiplying the exponents instead of adding them. Remember: when you see radicals being multiplied, immediately convert to fractional exponents. This transforms a potentially confusing radical problem into straightforward exponent arithmetic that follows familiar rules.

Question 22

A bus travels at a constant speed of 45 miles per hour. Let $t$ be time in hours and $d$ be distance in miles. If this relationship is graphed, which equation represents it?

$d = 45t$ (correct answer)
$t = 45d$
$d = 45 + t$
$d = t - 45$

Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. When creating an equation from a context, first identify the two quantities that are related (like distance and time), choose variables to represent them (like d for distance and t for time), then write an equation that captures how one depends on the other. From the context, distance is speed of 45 mph times time, we identify that d depends on t; the rate is 45 (that becomes our coefficient), and there's no starting distance (constant is 0), so the equation is d = 45t, which lets us calculate distance for any time! Choice C is correct because it accurately represents the relationship with the speed as the coefficient in a proportional equation. Choice A adds incorrectly: it uses d = 45 + t, but the context is constant speed meaning multiplication, not addition; 'per hour' means multiply by time! Quick trick: the words 'at a constant speed of' usually mean multiply rate by time (d = rt); listen to those cues! After you create your equation, test it with simple values: try t=1 (d=45 miles) and t=2 (d=90); if it matches the speed, you've nailed it!

Question 23

A student earns $9 per hour babysitting. Let$ h $be hours worked and$ E$ be earnings (in dollars). How should the coordinate plane be set up to show this context?

x-axis: Earnings ($), y-axis: Time (hours)
x-axis: Time (hours), y-axis: Earnings ($) (correct answer)
x-axis: $E$ (hours), y-axis: $h$ ($)
x-axis: Babysitting, y-axis: Money

Explanation: This question tests your ability to create equations from real-world relationships and set up appropriate graphs to visualize them. The coordinate plane helps us visualize relationships: the horizontal x-axis typically shows the independent variable (the one you choose or that changes first, like time or quantity), while the vertical y-axis shows the dependent variable (the one that responds, like cost or height). For this relationship, the x-axis should represent Time (hours) with that label, and the y-axis should represent Earnings ($) with that label. This makes sense because you choose how many hours to work (independent), and your earnings depend on that choice (dependent). Choice B is correct because it sets up the axes appropriately with the independent variable (time in hours) on the x-axis and the dependent variable (earnings in dollars) on the y-axis, including proper units. Choice A has the variables switched: it puts Earnings on the x-axis and Time on the y-axis, but remember—the independent variable (the one you start with or control) goes on the x-axis, and the dependent variable (the one that responds) goes on the y-axis. Remember the difference between independent and dependent variables: the independent variable is the one you can choose or control (like how many hours you work), and the dependent variable is the one that responds to your choice (like how much money you earn). Independent goes on the x-axis, dependent on the y-axis—this is the standard convention!

Question 24

Show that the equations $x^2+4x-5=0$ and $(x+2)^2=9$ have the same solutions by solving using the completed-square form.

They share solutions $x=1$ and $x=-5$ . (correct answer)
They share solutions $x=-1$ and $x=5$ .
They share the single solution $x=-2$ .
They have no real solutions.

Explanation: This question tests your ability to use completing the square to solve quadratic equations by transforming them into the form (x - p)² = q, which makes finding solutions straightforward by taking square roots. Completing the square is a method that transforms any quadratic equation ax² + bx + c = 0 into an equivalent equation (x - p)² = q that has the same solutions but is much easier to solve: once in this form, we just take the square root of both sides (remembering ±!), giving x - p = ±√q, then solve for x = p ± √q. This method works for ALL quadratics, even ones that don't factor nicely! After completing the square to get (x + 2)² = 9, we can see: since 9 is positive, we can take its square root, giving two real solutions. From (x + 2)² = 9, we get x + 2 = ±3, so x = -2 + 3 = 1 or x = -2 - 3 = -5. Now let's solve x² + 4x - 5 = 0: move constant to right: x² + 4x = 5, add (4/2)² = 4: x² + 4x + 4 = 9, factor: (x + 2)² = 9, solve: x = -2 ± 3, giving x = 1 or x = -5. The value of q immediately tells us how many real solutions exist! Choice A correctly identifies that both equations share solutions x = 1 and x = -5 with accurate arithmetic and proper form. Choice B has the signs reversed: the solutions are x = 1 and x = -5, not x = -1 and x = 5. Watch those signs! When we have (x + 2)² = 9 and take square roots, we get x + 2 = ±3, leading to x = -2 + 3 = 1 or x = -2 - 3 = -5. To verify your completing-the-square work: expand your (x - p)² + k form back out using FOIL, and you should get back to your original quadratic. If you don't, there's an error. Also, solve the original equation using the quadratic formula—you should get the same solutions. These double-checks catch mistakes and build confidence!

Question 25

Rewrite $8^t$ in terms of a power of 2 using exponent properties.

$2^{t+3}$
$6^t$
$2^{t/3}$
$2^{3t}$ (correct answer)

Explanation: This question tests your ability to use exponent properties to transform exponential expressions into equivalent forms that reveal information like interest rates at different time scales (annual, monthly, quarterly, etc.). The key insight is recognizing that 8 = 2^3, which allows us to rewrite 8^t in terms of powers of 2. This transformation uses the property that when you have a power raised to another power, you multiply the exponents. To simplify 8^t: recognize 8 = 2^3, so 8^t = (2^3)^t = 2^(3t). The exponent properties let us rewrite expressions with different bases when one base is a power of another. Choice B correctly rewrites 8^t as 2^(3t) using the fact that 8 = 2^3 and applying the power-of-a-power property. Choice A uses the wrong exponent property: it seems to apply something like 8^t = 2^(t+3), but this doesn't follow from any valid exponent rule. Check: when t=1, 8^1 = 8, but 2^(1+3) = 2^4 = 16, not 8! Always verify your transformation produces an equivalent expression. The exponent properties work because exponents represent repeated multiplication: 8 = 2^3 means 2·2·2. So 8^t = (2·2·2)^t, and when you have t factors of (2·2·2), that's the same as having 3t factors of 2, which equals 2^(3t). These properties aren't arbitrary—they follow from what exponents fundamentally mean!

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