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The quadratic function is graphed on a coordinate plane. What is the vertex of the quadratic function ?
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Algebra
Practice Test 44 for Algebra: real questions and explanations from the Varsity Tutors practice-test pool.
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Question 1 of 25
The quadratic function g(x)=(x−1)2−9 is graphed on a coordinate plane. What is the vertex of the quadratic function g(x)=(x−1)2−9?
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The quadratic function g(x)=(x−1)2−9 is graphed on a coordinate plane. What is the vertex of the quadratic function g(x)=(x−1)2−9?
Explanation: This question tests your understanding of how to identify key features of quadratic functions and their graphs, specifically the vertex. For a quadratic function in the form y = a(x - h)² + k, the vertex is the point (h, k), which is the highest point if the parabola opens down (a < 0) or the lowest point if it opens up (a > 0). This quadratic is in vertex form y = a(x - h)² + k, where we can read the vertex directly: comparing g(x) = (x - 1)² - 9 to the standard form, we have a = 1, h = 1, and k = -9, so the vertex is (1, -9). The vertex is the turning point—the very top or very bottom of the parabola! Choice C is correct because it properly identifies the vertex as (1, -9) by reading directly from the vertex form. Great job identifying this! Choice A gives (1, 9) instead of (1, -9), making a sign error with the k-value. In the form (x - 1)² - 9, the -9 is the k-value, not +9. We all make sign errors—that's why checking our work is so important! With quadratics in vertex form y = a(x - h)² + k, the vertex is right in front of you: it's (h, k). Watch out though—there's a minus sign in the form, so if you see (x - 3)², the h-value is positive 3, but if you see (x + 3)², the h-value is -3!
A gym charges a \25sign−upfeeplus$10permonth.LetmbethenumberofmonthsandC(m)bethetotalcost(indollars).WritethelinearfunctionC(m)$.
Explanation: This question tests your ability to construct linear functions from given information like descriptions. From context descriptions, listen for the clues: 'per,' 'each,' or 'constant rate' suggest linear (use that rate as slope), while 'percent growth,' 'doubles,' or 'halves' suggest exponential. 'Starting with' or 'initial' tells you the y-intercept (linear) or initial value (exponential). The gym charges a 25sign−upfee(thisisoury−intercept,thecostwhenm=0)plus10 per month (this is our slope, the rate of change). So C(m) = 10m + 25. Choice B correctly constructs C(m) = 10m + 25 by identifying the monthly rate as the slope (10) and the sign-up fee as the y-intercept (25). Perfect! Choice A incorrectly swaps these values - remember, the 'per month' rate is what multiplies the variable, while the one-time fee is the constant term. Context tip: if the problem says 'starts at [value],' that's your y-intercept. If it says 'increases by [number] each time,' that's slope. The language tells you exactly what you need!
Given the function f(x)=2x−3, what is f(4)?
Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. Function notation f(x) tells us the rule for calculating outputs from inputs: when you see f(4), it means 'substitute 4 for every x in the function formula,' like filling in a blank everywhere you see x. Starting with f(x) = 2x - 3 and finding f(4), we substitute 4 for x everywhere: f(4) = 2(4) - 3. Now we calculate step by step: 2(4) - 3 = 8 - 3 = 5. Choice A is correct because it properly substitutes 4 for x in the function and calculates accurately: 2(4) - 3 = 5. Nice work if you got this! Choice B has the right idea but makes an arithmetic error, calculating 2(4) + 3 = 11 instead of 2(4) - 3 = 5. Double-checking your arithmetic is always a good idea! Here's the foolproof way to evaluate functions: (1) write out the function formula, (2) wherever you see the variable, write the input value in parentheses, (3) calculate step by step using order of operations.
Using the quadratic formula, solve x2+3x−10=0.
Explanation: This question tests your ability to solve quadratic equations using the quadratic formula, which is an essential skill in Algebra 1. The quadratic formula x = (-b ± √(b² - 4ac))/(2a) works for ANY quadratic equation ax² + bx + c = 0—it's your reliable backup method when other approaches don't work easily, and it also reveals when you have complex solutions (when the part under the square root, b² - 4ac, is negative). Using the quadratic formula with a = 1, b = 3, c = -10: x = (-3 ± √(3² - 4(1)(-10)))/(2·1) = (-3 ± √(9 + 40))/2 = (-3 ± √49)/2 = (-3 ± 7)/2. This gives x = (-3 + 7)/2 = 4/2 = 2 and x = (-3 - 7)/2 = -10/2 = -5. The ± in the formula is what gives us two solutions! Choice A is correct because it properly applies the method and includes both solutions with correct arithmetic. Well done! Choice B has a sign error in the quadratic formula: the formula starts with -b (negative b), so with b = 3, we use -3, not +3, leading to the wrong signs in the solutions. Those signs are tricky! For the quadratic formula, here's a trick: write out 'x = (-b ± √(b² - 4ac))/(2a)' before you start, then carefully substitute a, b, and c from your equation. If you try to do it in your head, it's easy to drop a negative sign or mix up the numbers. Quick check: after solving, substitute your answers back into the original equation. If you get 0 = 0, great! If not, you made an error somewhere. This catch-your-own-mistakes habit is one of the best math skills you can develop!
Transform and solve by completing the square: x2+8x+7=0. Write it as (x−p)2=q and then find x.
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. The completing-the-square process creates a perfect square trinomial on one side: for x² + bx, we add (b/2)²—half the middle coefficient, squared—to both sides. This turns x² + bx + (b/2)² into (x + b/2)², a perfect square that factors as a binomial squared. Then we solve by taking square roots. It's like setting up a problem in a form where the answer is obvious! Solving x² + 8x + 7 = 0 by completing the square: (1) Move constant to right: x² + 8x = -7. (2) Take half of 8 to get 4, square it to get 16, add to both sides: x² + 8x + 16 = -7 + 16. (3) Left side is perfect square (x + 4)², right side simplifies to 9: (x + 4)² = 9. (4) Take square roots: x + 4 = ±3. (5) Solve: x = -4 ± 3, so x = -1 or x = -7. Done! Choice A correctly completes the square to get (x + 4)² = 9 and solves to get x = -4 ± 3 with accurate arithmetic and proper form. Choice B calculates (b/2)² incorrectly: half of 8 is 4, and squaring gives 16, not 64. This perfect square term is the heart of completing the square—getting it right is crucial! An error here throws off everything that follows. The (b/2)² trick: if you have x² + 6x, half of 6 is 3, and 3² = 9, so you add 9 to complete the square, getting (x + 3)². If you have x² - 10x, half of -10 is -5, and (-5)² = 25 (positive!), so add 25 to get (x - 5)². The sign of b affects the sign in the binomial, but (b/2)² is always positive. Practice this on a few examples and it becomes automatic!
For which of the following quadratic equations would factoring be the least efficient solution method?
Explanation: Choice A factors as (x−3)(x−4)=0. Choice C is a difference of squares: (x−4)(x+4)=0. Choice D can be simplified by factoring out 2: 2(x2−4x+4)=2(x−2)2=0. Choice B has the form 3x2+7x−2=0. To factor this, we need two numbers that multiply to (3)(−2)=−6 and add to 7. These would be −1 and 6, but the factoring process with the coefficient 3 makes this more complex than using the quadratic formula directly.
A movie theater charges 8foraticketplus3 for each snack. Let C be the total cost (in dollars) and let s be the number of snacks. What equation represents the relationship between C and s?
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 cost and number of items), choose variables to represent them (like C for cost and n for number), then write an equation that captures how one depends on the other. From the context, the theater charges 8foraticket(afixedcost)plus3 for each snack (a variable cost), we identify that total cost C depends on number of snacks s. The fixed ticket cost is 8(that′sourconstantterm),andtheratepersnackis3 (that becomes our coefficient for s). So the equation is C = 8 + 3s. This equation lets us calculate the total cost for any number of snacks! Choice B is correct because it accurately represents the relationship with 8asthefixedticketcostand3s as the cost that varies with the number of snacks. Choice A has the same numbers but switches their roles: it makes the snack cost 8eachandtheticketonly3, but remember—the problem says the ticket is 8andeachsnackis3. When translating words to equations, make sure each part of the equation corresponds to something in the description! Quick trick: the words in the problem often tell you what operation to use. 'Per' or 'each' usually means multiply (like 3persnack=3timesnumberofsnacks).′Plus′or′and′meansadd(like8 ticket plus snack cost). Listen to the language!
Use structure to rewrite x2+10x+25 as a single squared binomial.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²: the first and last terms are perfect squares, and the middle term is exactly twice their product. Recognizing this pattern lets you write the trinomial as a single squared binomial! Let's check if x² + 10x + 25 is a perfect square trinomial. First term: x² is clearly a perfect square. Last term: 25 = 5², also a perfect square. Middle term: 10x should equal 2·x·5 = 10x. Yes! All three conditions are met, and all signs are positive, so this follows the (a + b)² pattern where a = x and b = 5. Therefore, x² + 10x + 25 = (x + 5)². Choice C correctly identifies this as (x + 5)². Excellent pattern recognition! Choice A would give us (x + 5)(x - 5) = x² - 25, which is a difference of squares with no middle term—not what we started with. For perfect square trinomials, check three things: (1) First and last terms are perfect squares, (2) Middle term has the right coefficient (twice the product of what you're squaring), (3) Signs match the pattern (++ means (a+b)², +- means (a-b)²). Example: x² + 10x + 25 has x² (✓), 25 = 5² (✓), and 10x = 2·x·5 (✓), so it's (x + 5)²!
A piecewise function p(x) is defined such that p(2)=5, p(4)=3, and the function decreases linearly from x=2 to x=4. For x<2, p(x)=x+7. What is the value of p(2)+p(−2)?
Explanation: Given p(2) = 5. For p(-2), since -2 < 2, use p(x) = √(x+7). So p(-2) = √(-2+7) = √5. Therefore p(2) + p(-2) = 5 + √5. Choice B would require √5 = 3, which is incorrect. Choice C uses √7 instead of √5. Choice D assumes p(-2) = 5, which is incorrect.
Consider the recursively defined sequence g(n)=3g(n−1)−g(n−2) for n≥3, with initial terms g(1)=2 and g(2)=7. Which statement about this sequence is true?
Explanation: To determine which statement is true, we need to calculate a few terms and analyze the sequence properties. g(1) = 2, g(2) = 7, g(3) = 3(7) - 2 = 19, g(4) = 3(19) - 7 = 50. Choice A is wrong because differences are: 7-2=5, 19-7=12, 50-19=31 (not constant). Choice B is wrong because ratios are: 7/2=3.5, 19/7≈2.7, 50/19≈2.6 (not constant). Choice D is wrong because the sequence is increasing: 2 < 7 < 19 < 50. Choice C is correct because sequences are functions whose domain is a subset of the integers, and here the domain is {1, 2, 3, 4, ...}, which is the set of positive integers.
Subtract and simplify (the difference of two rational expressions is a rational expression): x−2x−x−23
Explanation: This question tests your understanding of how to subtract 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 x/(x-2) - 3/(x-2), we're lucky—the denominators are already the same! Just like 5/7 - 2/7 = 3/7, we subtract the numerators and keep the denominator: x/(x-2) - 3/(x-2) = (x-3)/(x-2). Choice A correctly subtracts the numerators to get (x-3)/(x-2) since the denominators were already the same. Great work! Choice B incorrectly gets x-1 in the numerator, C has the wrong denominator, and D shows the original expression without simplifying. Common mistake: trying to cancel before getting common denominator in addition. You can only cancel FACTORS (things being multiplied), not TERMS (things being added). So in x/(x-2) - 3/(x-2), you cannot cancel anything—you can only subtract numerators because denominators are already the same: (x-3)/(x-2). When denominators match, it's just like regular fraction subtraction!
Transform and solve: x2−6x+2=0. Use completing the square to write it as (x−p)2=q and then find x.
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! Solving x² - 6x + 2 = 0 by completing the square: (1) Move constant to right: x² - 6x = -2. (2) Take half of -6 to get -3, square it to get 9, add to both sides: x² - 6x + 9 = -2 + 9. (3) Left side is perfect square (x - 3)², right side simplifies to 7: (x - 3)² = 7. (4) Take square roots: x - 3 = ±√7. (5) Solve: x = 3 ± √7. Done! Choice A correctly completes the square to get (x - 3)² = 7 and solves to get x = 3 ± √7 with accurate arithmetic and proper form. Choice C makes an arithmetic error when simplifying the right side: -2 + 9 = 7, not 11, so we should get (x - 3)² = 7, not (x - 3)² = 11. Always double-check your arithmetic when combining terms on the right side after adding (b/2)² to both sides! Why this works: when you add (b/2)² to x² + bx, you're creating the expansion of (x + b/2)²: remember (a + b)² = a² + 2ab + b², so (x + b/2)² = x² + 2(x)(b/2) + (b/2)² = x² + bx + (b/2)². That's why adding (b/2)² makes a perfect square! It's not random—it completes the perfect square trinomial pattern.
A city wants to describe how busy a bus stop is during the afternoon. Which quantity is most relevant for a descriptive model of the stop’s activity?
Explanation: This question tests your ability to identify and define appropriate quantities for mathematical modeling—deciding what to measure, how to measure it, and what units to use to describe a real-world situation quantitatively. Relevant quantities are those that actually affect or describe the aspect you're modeling: if modeling a basketball team's scoring ability, 'points per game' and 'shooting percentage' are relevant, but 'jersey numbers' and 'player heights' are less relevant (heights might matter for some analyses, but not for scoring specifically). Always ask: does this quantity help describe what I'm trying to understand? If no, it's irrelevant clutter. Evaluating which quantities are relevant for describing bus stop activity: Number of passengers who board per 10-minute interval: relevant because it directly measures how many people use the stop—the core of 'activity.' Favorite music genre: irrelevant because musical preferences don't affect or describe how busy the stop is. Color of bus seats: irrelevant because seat color doesn't change with activity levels. Driver's middle name: irrelevant because it has no connection to stop usage patterns. The key is asking: does this quantity help us understand or describe how busy the bus stop is? Only passenger counts do. Choice A correctly identifies the relevant quantity that directly measures bus stop activity—passenger boarding counts capture exactly how busy the stop is. Choice B includes an irrelevant quantity: while passengers' music preferences are interesting, they don't actually affect or describe how busy the bus stop is. For example, knowing 40% prefer rock music tells us nothing about whether 10 or 100 people use the stop. Including irrelevant quantities clutters the model without adding understanding—keep only what matters for the specific modeling goal! Relevance is purpose-dependent: when modeling 'student academic performance,' test scores and attendance are relevant, but student height is irrelevant (for academic performance specifically—height might be relevant for modeling basketball performance!). Always ask: relevant for what purpose? The same situation can be modeled different ways depending on what aspect you're trying to describe!
The quadratic function h(t)=−16t2+64t represents the height of a ball thrown upward, where t is time in seconds and h(t) is height in feet. By factoring to find the zeros, determine when the ball returns to the ground and what is the total time the ball is in the air.
Explanation: When you encounter a quadratic function representing projectile motion, you're looking for when the object hits the ground—which occurs when the height equals zero. Finding the zeros through factoring will give you the complete picture of the ball's flight. To find when h(t)=0, set the equation equal to zero: −16t2+64t=0. Factor out the common term −16t: −16t(t−4)=0. Using the zero product property, either −16t=0 or t−4=0, giving you t=0 and t=4. These zeros tell the complete story: t=0 is when the ball is thrown (starts at ground level), and t=4 is when it returns to the ground. The total air time is 4−0=4 seconds. Choice A incorrectly uses the coefficient 16 from the equation as the time value, confusing the mathematical constants with the physical meaning. Choice B likely comes from incorrectly factoring or misunderstanding that you need both zeros to determine the flight duration. Choice C might result from calculation errors when factoring, perhaps confusing the vertex time (when the ball reaches maximum height) with the landing time. The correct answer is D: the ball returns to the ground at t=4 seconds, with a total air time of 4 seconds. Strategy tip: For projectile motion problems, always factor completely to find both zeros—they represent the start and end of the flight, giving you the total time in the air.
The function f(x)=−(x−2)2+5 is in vertex form. What does this reveal about the graph?
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. The three forms of a quadratic each reveal different features: standard form f(x) = ax² + bx + c shows the y-intercept (c) clearly; factored form f(x) = a(x - p)(x - q) shows the zeros (p and q); vertex form f(x) = a(x - h)² + k shows the vertex (h, k) and extreme value (k). Knowing how to convert between forms lets you see whichever features you need! We can analyze this quadratic using both methods: The given vertex form f(x) = -(x - 2)² + 5 directly reveals the vertex at (2, 5). Notice how the axis of symmetry x = 2 from the vertex would match (p + q)/2 if factored—both methods find the same axis because it's a property of the parabola! Since a = -1 < 0, it's a maximum at 5. Choice A correctly identifies from vertex form the vertex at (2,5), axis at x=2, and maximum value 5. Choice C identifies the vertex correctly but confuses maximum with minimum: since a = -1 is negative, the parabola opens down, making the vertex a maximum, not a minimum. The sign of a determines whether the vertex is the highest or lowest point! The three forms, three features connection: Standard form (ax² + bx + c) → see y-intercept c immediately. Factored form (a(x-p)(x-q)) → see zeros p, q immediately. Vertex form (a(x-h)²+k) → see vertex (h,k) immediately. Each form is optimized to show certain features! Convert to the form that shows what you need.
Solve and check for extraneous solutions:
x−3x+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 - 3) and our solution is x = 3, 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 x−3x+1=2: (1) Multiply both sides by (x−3): x+1=2(x−3). (2) Distribute: x+1=2x−6. (3) Solve: 1=x−6, so x=7. (4) Check in original: Does x=7 make the denominator zero? 7−3=4=0. Good! Verify it satisfies equation: 7−37+1=48=2. ✓ Final answer: x=7. Choice B correctly solves to get x=7 and verifies it doesn't make the denominator zero, confirming it's a valid solution. Choice C suggests x=3 is valid, but checking: x=3 makes the denominator 3−3=0, which means division by zero—undefined! This is exactly what makes a solution extraneous. Always verify: for rational equations, does the solution make any denominator zero? If yes, reject it! 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. Extraneous solution red flags: For rational equations, if your solution is a value that appeared in any denominator, CHECK IT CAREFULLY—it's likely extraneous!
A student is buying notebooks and pens. Let x be the number of notebooks and y be the number of pens.
Which statement is true about the point (x,y)=(3.5,6)?
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! To check if (3.5, 6) is viable, we substitute into each constraint: for 4x + y ≤ 25, 4(3.5) + 6 = 14 + 6 = 20 ≤ 25 (true); for x + y ≥ 8, 3.5 + 6 = 9.5 ≥ 8 (true); for x ≥ 0, y ≥ 0 (true). But checking context: must be whole numbers, and 3.5 is fractional, which is impossible for items. Conclusion: nonviable because violates whole-number requirement. Choice C is correct because it correctly identifies that it violates the whole-number requirement even though it satisfies the inequalities. Choice B says it's nonviable but incorrectly checks the budget: 20 ≤ 25 is satisfied, not violated. Remember, mathematical satisfaction isn't enough—context matters! 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. Also do a reality check: negative quantities? fractional items when must be whole? These context violations also make solutions nonviable! Remember: viable means 'it could actually happen in real life.' Mathematically satisfying inequalities is necessary, but not sufficient. Ask yourself: In this context, can quantities be negative? Can they be fractions? Are there other real-world restrictions? A solution can satisfy all the math but still be nonviable if it violates reality!
Two students are analyzing the sequence defined by v(1)=2, v(2)=5, and v(n)=v(n−1)+v(n−2)+1 for n≥3. Student A claims the sequence grows faster than the Fibonacci sequence, while Student B claims it grows at the same rate. Who is correct and why?
Explanation: When comparing recursive sequences, you need to examine how the recursive formula affects the growth pattern, not just the pattern itself. Let's trace through both sequences to see what happens. The given sequence follows v(n)=v(n−1)+v(n−2)+1 with v(1)=2,v(2)=5. The Fibonacci sequence follows F(n)=F(n−1)+F(n−2) with F(1)=1,F(2)=1. Computing the first few terms: Given sequence: 2, 5, 8, 14, 23, 38, 62... Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34... Notice that at each step, the given sequence adds an extra 1 beyond what Fibonacci-style growth would produce. This extra 1 at every step accumulates over time, creating a compounding effect that makes each term progressively larger relative to Fibonacci. Answer choice A is wrong because having the same recursive pattern doesn't guarantee the same growth rate when there are additional terms. Choice B incorrectly assumes that adding 1 is negligible—but when you add 1 at every step in a recursive sequence, those additions compound and significantly impact growth. Choice C focuses on initial values, but even if we started with the same initial values, the "+1" term would still make this sequence grow faster. Choice D correctly identifies that the "+1" term is the key difference, making each value larger than what pure Fibonacci growth would produce. Remember: In recursive sequences, even small additional terms compound over time and can dramatically change growth rates compared to their base patterns.
Perform the indicated operation and simplify (the product is a rational expression): x3⋅x+2x2
Explanation: This question tests your understanding of how to multiply rational expressions—algebraic fractions that work just like regular fractions but with variables. Multiplying rational expressions works just like multiplying numeric fractions: multiply the numerators together and multiply the denominators together, giving (a/b)·(c/d) = (ac)/(bd). But here's the smart way: factor first, cancel common factors, THEN multiply—it keeps the numbers smaller and the result already simplified! Let's multiply: (3/x) · (x²/(x+2)) = (3·x²)/(x·(x+2)) = 3x²/[x(x+2)]. Now we can simplify by canceling one factor of x from numerator and denominator: 3x²/[x(x+2)] = 3x/(x+2). Choice C correctly multiplies and simplifies to 3x/(x+2) by multiplying numerators and denominators, then canceling the common factor of x. Great work! Choice A shows the unsimplified form, choice B cancels too many x's (you can only cancel one!), and choice D has the wrong denominator. The golden rule for multiplying and dividing rationals: factor everything you can BEFORE you multiply or cancel. This prevents working with huge expressions and catches opportunities to simplify. Example: [(x²-4)/x]·[x/(x+2)] looks messy, but factor x²-4 to get [(x+2)(x-2)/x]·[x/(x+2)], cancel the (x+2) and x, leaving just (x-2). So much cleaner!
How many terms are in the expression −3(x−2)(x+1)?
Explanation: This question tests your understanding of the parts of algebraic expressions—specifically, how to identify terms. Terms are the parts of an expression that are added or subtracted from each other: think of them as the separate 'chunks' connected by + or - signs. For example, in 3x² - 5x + 2, there are three terms: 3x², -5x, and 2 (notice that the -5x includes the minus sign). Let's look at the expression -3(x - 2)(x + 1) and find the terms. There are no + or - signs at the main level separating parts: it's all multiplied together as one big chunk. That gives us 1 term total. Choice C is correct because it properly identifies 1 term, following the definition that terms are separated by + or -. You've got it! Choice A counts 3, but let me help clarify: the parts inside are factors being multiplied, not terms being added. It's an easy mistake to make when you're learning to identify these parts! Here's an easy way to identify terms: look for the + and - signs that aren't inside parentheses—those are your term separators! Everything between those signs (including the sign right before it) is one term. So in -3(x - 2)(x + 1), there are no top-level + or -, so it's just one term!
For the function f(x)=2x2+1, what is f(−3)?
Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. Function notation f(x) tells us the rule for calculating outputs from inputs: when you see f(-3), it means 'substitute -3 for every x in the function formula,' like filling in a blank everywhere you see x. When evaluating with a negative number like f(-3), we need to be extra careful with signs! Substituting -3 for x in f(x) = 2x² + 1, we get 2(-3)² + 1 = 2(9) + 1 = 18 + 1 = 19. Notice how the parentheses around -3 help keep track of the negative! Choice B is correct because it properly substitutes -3 for x in the function and calculates accurately: 2(9) + 1 = 19. Nice work if you got this! Choice D is a common slip-up: it makes a sign error when working with the negative input -3. When you substitute a negative number, use parentheses to keep track: (-3)² is positive, but forgetting parentheses can lead to the wrong sign. When substituting negative numbers, always use parentheses to protect yourself from sign errors: write f(-3) = 2(-3)² + 1, not f(-3) = 2-3² + 1. The parentheses keep everything clear!
What is the boundary line for the inequality 2x+y≥6?
Explanation: This question tests your understanding of graphing linear inequalities and how the solution is represented as a shaded half-plane on the coordinate plane. The boundary line for an inequality is the line you'd get if you changed the inequality to equals: for 2x + y > 6, the boundary is 2x + y = 6. The line is dashed for strict inequalities (< or >) because points ON the line don't satisfy the inequality, and solid for ≤ or ≥ because boundary points ARE solutions. For 2x + y ≥ 6, the boundary is simply 2x + y = 6, which you can graph by finding intercepts (x=3 when y=0, y=6 when x=0) and drawing a solid line since ≥ includes equality. Choice B correctly identifies the boundary as 2x + y = 6 because it's the equality version of the inequality. Choice A includes the inequality symbol, but remember, the boundary is just the line itself—nice try, but focus on replacing the inequality with equals. The solid-or-dashed rule is simple: if you see ≤ or ≥ (the inequality has a line underneath showing 'or equal to'), make the boundary line solid because those points are included. If you see < or > (strict inequality, no line underneath), make it dashed because boundary points don't count. Think: the line under the inequality symbol = solid line on the graph!
If f(x)=x2 and g(x)=2x−1, which expression represents f(g(x))?
Explanation: This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. The notation f(g(x)) can look intimidating, but it's just telling you the order: work from the inside out, just like nested parentheses in arithmetic. If f(x) = x² and g(x) = 2x - 1, then f(g(x)) means 'plug 2x - 1 into f,' giving f(2x - 1) = (2x - 1)². You're replacing every x in f with the entire expression for g(x)! To find f(g(x)) where f(x) = x² and g(x) = 2x - 1, we substitute the entire expression for g(x) into f: f(g(x)) = f(2x - 1) = (2x - 1)². Every place you see x in f, you replace it with 2x - 1—and since we're squaring, we need parentheses around the whole thing! Choice A correctly composes the functions by properly substituting g into f, giving (2x - 1)². Choices B and C make a substitution error: when replacing x with 2x - 1 in f(x) = x², we need to substitute the ENTIRE expression for g(x), including handling it as a grouped quantity. Since f(x) = x², then f(2x - 1) = (2x - 1)², not 2x² - 1! To compose functions into a formula: (1) Write down the inner function's expression g(x) = 2x - 1, (2) In the outer function f(x) = x², everywhere you see x, replace it with (2x - 1)—treat it as a single chunk, use parentheses!, (3) The result is (2x - 1)², which we could expand if needed.
Factor f(x)=x2−9x+20 to find the zeros and the axis of symmetry of the graph.
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! To find the zeros of f(x) = x² - 9x + 20, we factor: looking for two numbers that multiply to 20 and add to -9, we find -4 and -5. So f(x) = (x - 4)(x - 5). Setting each factor to zero: x - 4 = 0 gives x = 4, and x - 5 = 0 gives x = 5. These are our zeros! The axis of symmetry is at x = (4 + 5)/2 = 9/2, exactly halfway between the zeros. Choice A correctly factors to get (x-4)(x-5) showing zeros at x=4,5 and axis at x=9/2. Choice B has a sign error in the factoring: it lists zeros as x=-4 and x=-5, but since the sum of roots is 9 (positive) and product 20 (positive), both zeros should be positive—remember, the signs in the factors determine the root signs! Feature-finding strategy: (1) Need zeros? Factor into (x - p)(x - q) form and set factors = 0. (2) Need vertex? Complete the square to get (x - h)² + k form and read (h, k). (3) Need axis of symmetry? Use x = h from vertex OR x = (p + q)/2 from zeros OR x = -b/(2a) from standard form—all three work! (4) Need extreme value? It's k from vertex form. Choose the right tool for what you need!
Function A is given by A(t)=50(1.10)t.
Function B is described verbally as: “an exponential function with initial value 50 and growth factor 1.05 per unit of t.”
Which function has the faster growth rate?
Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare growth rates, note A's formula has growth factor 1.10; B's verbal description gives 1.05, so 1.10 > 1.05 means A grows faster. Choice B correctly identifies that function A has the faster growth rate because 1.10 > 1.05. It's common to confuse which factor is larger, but remember the bigger the base >1, the faster the exponential growth—keep practicing! Know what each representation shows best: formulas are great for calculating specific values and seeing patterns in the equation; graphs excel at showing maximums, minimums, and overall shape; tables are perfect for finding exact values at specific points; descriptions summarize key features—use each representation's strengths! Quick y-intercept trick: in a formula, set x=0 and calculate; on a graph, see where it crosses the y-axis; in a table, find the y-value when x=0—three different methods, same property!