Question 1 of 25
A student graphs the equation and notices the graph forms a V-shape. When asked to explain why the point appears on the graph, which explanation demonstrates the best understanding?
Opening subject page...
Loading your content
Algebra
Practice Test 47 for Algebra: real questions and explanations from the Varsity Tutors practice-test pool.
0%
0 / 25 answered
Question 1 of 25
A student graphs the equation y=∣x−2∣ and notices the graph forms a V-shape. When asked to explain why the point (5,3) appears on the graph, which explanation demonstrates the best understanding?
Question Navigator
A student graphs the equation y=∣x−2∣ and notices the graph forms a V-shape. When asked to explain why the point (5,3) appears on the graph, which explanation demonstrates the best understanding?
Explanation: When you encounter absolute value functions, remember that to verify if a point lies on the graph, you substitute the x-coordinate into the equation and check if it produces the given y-coordinate. For the equation y=∣x−2∣, let's test whether (5,3) is actually on the graph. Substituting x=5: y=∣5−2∣=∣3∣=3. Since this gives us the point (5,3), the point is indeed on the graph. This direct substitution method is the fundamental way to verify any point on any function. Looking at the wrong answers: Choice B incorrectly assumes that having positive coordinates automatically means a point fits the V-shape, but this ignores the specific equation entirely—many positive coordinate points don't lie on this particular graph. Choice C mentions being "3 units from the vertex," but this is imprecise and doesn't demonstrate understanding of how absolute value functions work. The vertex is at (2,0), and while there are distance relationships, that's not how we verify points on graphs. Choice D applies the distance formula between (5,3) and (2,0), which calculates the distance between two points but has nothing to do with whether a point satisfies the equation. Choice A demonstrates the correct mathematical reasoning by using direct substitution to verify the relationship between x and y coordinates. Remember: To check if any point lies on a graph, always substitute the x-value into the equation and see if you get the corresponding y-value. This works for linear, quadratic, absolute value, and all other types of functions.
An investment account is modeled by A(t)=1000(1.05)t, where t is in years. What does the base 1.05 represent?
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. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function A(t) = 1000(1.05)^t, the base is 1.05. To find the percent rate, we calculate r = 1.05 - 1 = 0.05. Converting to percent: 0.05 × 100% = 5%. Since 1.05 is greater than 1, this is growth, specifically 5% growth per year. Choice C correctly identifies the base 1.05 as a growth factor representing 5% annual growth by showing correct reasoning. Excellent! Choice D makes a common percent mistake: the base 1.05 doesn't mean 105% growth—it means 5% growth! The 1 represents 'what you already have' (100%), and the 0.05 is the additional 5%, for a total of 105% of the previous amount (which is 5% growth). Don't confuse the factor with the rate: if something grows by 5% per year, the growth RATE is 5% (r = 0.05), but the growth FACTOR is 1.05 (b = 1.05). Each year you have 105% of what you had (100% + 5%), which means multiplying by 1.05.
Complete the square to solve x2+8x+7=0.
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, giving x = -4 + 3 = -1 or x = -4 - 3 = -7. Done! Choice A correctly completes the square to get (x + 4)² = 9 and solves to get x = -4 ± 3, which gives x = -1 or x = -7, with accurate arithmetic and proper form. Choice C makes an arithmetic error when simplifying the right side: -7 + 16 = 9, not 7, so we should get (x + 4)² = 9, not (x + 4)² = 7. Completing the square involves several arithmetic steps, especially when combining terms on the right side. Take your time! The ± is crucial: from (x + 4)² = 9, taking square roots gives x + 4 = ±3 (both +3 and -3), so x = -4 + 3 = -1 OR x = -4 - 3 = -7. Two solutions! Don't forget the ± and don't forget to split it into two separate solutions. Check both: -1 and -7 both satisfy the original equation? Yes!
Determine whether the function f(x)=2x2−3 has a constant rate of change. (Recall: a constant rate of change means Δf/Δx is the same for all equal x-intervals.)
Explanation: This question tests your ability to recognize when a relationship has a constant rate of change—which is the defining characteristic of linear functions. Linear functions are the ONLY functions with constant rates of change: if a graph is a straight line, the rate is constant. If the graph curves (like a parabola or exponential curve), the rate is changing. You can visually spot constant rate—it's straightness! A steeper line has larger constant rate, flatter line has smaller constant rate, but both are constant as long as the line is straight. Looking at the function f(x) = 2x² - 3: This is a quadratic function (has x²), and quadratic functions have variable rates of change—the rate is different at different x-values, so it's not constant. To verify: at x = 0 to x = 1, Δf = f(1) - f(0) = -1 - (-3) = 2. At x = 1 to x = 2, Δf = f(2) - f(1) = 5 - (-1) = 6. The rates are different (2 vs 6), confirming non-constant rate. Choice B correctly identifies the rate as non-constant because the function is quadratic (nonlinear), which always has varying rates of change. Choice A confuses the coefficient of x² with the rate of change—the coefficient tells us about the parabola's shape, not whether the rate is constant. Only linear functions (y = mx + b form) have constant rates. Quadratics, exponentials, and other nonlinear functions all have rates that vary as x changes! Formula clue: if the function is y = mx + b (first degree, just x, not x² or 2^x or anything else), the rate is constant and equals m. Any other form (quadratic, exponential, rational, radical) has non-constant rate. The power of x tells you: power of 1 (or just x) = constant rate, any other power = non-constant rate.
The sequence is defined recursively by a1=4 and an+1=an+3. Translate this to an explicit formula for an.
Explanation: This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). The difference between recursive and explicit formulas: recursive is like climbing stairs one at a time (you need to know the previous term), while explicit is like taking an elevator directly to any floor (you can find the nth term without finding all the ones before it). Both describe the same sequence, just different approaches! Given starting formula type as recursive with a₁=4 and aₙ₊₁=aₙ+3, we identify: this shows arithmetic with a₁=4 and d=3 from the addition rule. To convert to explicit formula type, we use the translation process of plugging into aₙ = a₁ + (n-1)d. So the explicit formula is aₙ=4+3(n-1). Choice C is correct because it properly identifies the first term as a₁=4 and the common difference as d=3, giving the explicit formula aₙ=4+3(n-1). Great work! Choice A has the right idea but uses the wrong formula structure: for arithmetic sequences, the explicit formula needs aₙ = a₁ + (n-1)d, but this choice has aₙ=4·3^{n-1} which is geometric. Writing explicit formulas: for arithmetic, use aₙ = a₁ + (n - 1)d (start with first term, add the difference (n-1) times); for geometric, use aₙ = a₁·r^(n-1) (start with first term, multiply by ratio (n-1) times). The (n-1) appears because the first term already includes one application! To convert between forms: from recursive to explicit, identify a₁ and d (or r), then plug into the explicit formula. From explicit to recursive, read off a₁ and find d (coefficient of n after simplifying) or r (the base of the exponent).
In the formula for density, ρ=Vm, mass m is measured in grams (g) and volume V is measured in cubic centimeters (cm3). What are the units of density ρ?
Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Every formula has dimensional consistency: the units on the left must match the units on the right. For density = mass/volume, if mass is in grams and volume is in cm³, we can determine density's units by performing the division with units included. Checking units for ρ = m/V: Mass m has units g (grams), volume V has units cm³ (cubic centimeters). Performing the division: ρ = m/V means density units = g/cm³. This reads as 'grams per cubic centimeter'—it tells us how many grams of material fit in each cubic centimeter of space. The division of units works just like division of numbers! Choice C correctly shows g/cm³ as the units of density when mass is in grams and volume is in cubic centimeters. Choice A has the units inverted (cm³/g)—this would be 'volume per mass' or specific volume, not density! Remember: in a fraction, the numerator unit goes on top, denominator unit goes on bottom. Units help you solve problems even when you're not sure of the formula: think 'what units should density have?' Density measures how much mass fits in a given volume, so it should be mass per volume: mass/volume. This reasoning leads directly to g/cm³. The units almost tell you the formula!
An investment grows by 12% per year, modeled by A(t)=P(1.12)t where t is in years. Rewrite (1.12)t to reveal an equivalent monthly growth factor using exponent properties.
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.). To find a monthly rate from an annual rate, we use the fact that 12 months of monthly compounding should equal 1 year of annual: if annual factor is 1.12, the monthly factor b satisfies b^12 = 1.12, so b = (1.12)^(1/12), meaning the monthly growth factor. The exponent properties let us write this as (1.12)^t = ((1.12)^(1/12))^(12t), showing both the yearly and monthly perspectives! To convert the annual expression (1.12)^t to monthly, we use the power-of-a-power property: first, recognize that t years = 12t months. We want (something)^(12t). What's that something? It's (1.12)^(1/12), because ((1.12)^(1/12))^(12t) = (1.12)^((1/12)·12t) = (1.12)^t by the power-of-a-power rule. Choice B correctly transforms using (b^a)^c = b^(ac) with the monthly factor (1.12)^(1/12) raised to the power 12t months. Choice C uses the wrong approach: it divides 1.12 by 12 to get approximately 0.093, but the monthly factor isn't found by dividing the annual factor by 12. We need (1.12)^(1/12), which is the 12th root of 1.12, not 1.12 divided by 12. Division would give simple interest, but this is compound interest! The power-of-a-power property (b^a)^c = b^(ac) is your main tool for time-base conversion: to convert annual rate b^t to monthly, write it as ((b)^(1/12))^(12t)—take the 12th root of b for the monthly factor, then raise to 12t (12 months × t years). Check your work: the exponents multiply to give (1/12)·(12t) = t, confirming equivalence!
The value of a laptop depreciates over time. Its value is modeled by V(t)=1200(0.85)t, where t is years and V(t) is in dollars. What does 0.85 represent in this context?
Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In an exponential function like y = a·b^x, the parameter a is the initial value (what y equals when x = 0, since b^0 = 1), and the base b is the growth factor (if b > 1) or decay factor (if 0 < b < 1). If b = 1.05, that means multiplying by 1.05 each time, which is a 5% increase! In the function V(t) = 1200(0.85)^t, the 1200 is the initial value (starting laptop value of $1200), and the base 0.85 means the value is multiplied by 0.85 each year. Since 0.85 = 1 - 0.15, this represents a 15% decrease per year. We subtract 0.85 from 1 to find the decay rate: 1 - 0.85 = 0.15 = 15%. Choice B is correct because it properly identifies that 0.85 means the laptop keeps 85% of its value each year, which is equivalent to a 15% decrease per year. Perfect! Choice C gets the direction wrong, saying the value increases when actually it decreases. With exponential functions, if b > 1 it's growth (getting bigger), if 0 < b < 1 it's decay (getting smaller). Check whether your base is above or below 1! For exponential functions y = a·b^x: a is what you have at time zero (plug in x = 0 and you get a), and b tells you the multiplication factor each time period. To find the percent rate: subtract 1 from b if b > 1 (like 1.05 → 0.05 = 5% growth), or subtract b from 1 if b < 1 (like 0.95 → 1 - 0.95 = 0.05 = 5% decay). Quick check for exponential: if the base b = 0.85, think '1 minus 0.15, so that's 15% decay.' The distance from 1 is the rate, and whether it's above or below 1 tells you growth or decay!
A recipe uses 750 mL of water. Convert this amount to liters. (Use 1000 mL=1 L.)
Explanation: This question tests your understanding of how units help us solve problems correctly, verify our work, and communicate results clearly—units aren't just labels, they're essential tools for mathematical reasoning. Units in calculations work like variables: they multiply, divide, and cancel just like algebraic expressions. When you convert 750 mL to liters, divide by 1000 (since 1000 mL = 1 L). This dimensional analysis helps you set up conversions correctly: 750 mL × (1 L/1000 mL) = 750/1000 L = 0.75 L. The unit cancellation guides the calculation! Converting 750 mL to L: We need a conversion factor that has L in the numerator and mL in the denominator, so they cancel: 750 mL × (1 L/1000 mL) = 750/1000 L. The units cancel: mL × L/mL = L. Calculation: 750 ÷ 1000 = 0.75 L. The unit cancellation confirms we set up the conversion correctly! Choice B correctly converts milliliters to liters with proper unit cancellation resulting in 0.75 L. Choice A multiplies by 10 instead of dividing by 1000: this would be correct if converting from centiliters (cL) to liters, but not from milliliters. The prefix 'milli-' means 1/1000, so 1000 mL = 1 L, which means we divide by 1000 to convert mL to L. Always pay attention to metric prefixes! Unit conversion strategy: (1) Write the starting value with units, (2) Multiply by conversion factor(s) set up as fractions so unwanted units cancel: (wanted unit)/(starting unit), (3) Cancel units systematically—cross out units that appear in both numerator and denominator, (4) Verify final units match what you want, (5) Calculate the numbers. For metric conversions, remember: kilo- = 1000, centi- = 1/100, milli- = 1/1000!
For the function p(x)=−3x+4, calculate the average rate of change of p(x) over the interval [−2,2].
Explanation: This question tests your understanding of average rate of change, which is a super important concept connecting slope, functions, and real-world rates like speed or growth. For a linear function, the average rate of change is the same as the slope and doesn't depend on which interval you choose—the function changes at a constant rate everywhere. To find the average rate of change of p(x) = -3x + 4 from x = -2 to x = 2, we first evaluate at the endpoints: p(-2) = -3*(-2) + 4 = 10 and p(2) = -3*2 + 4 = -2. Then we use the formula: average rate = [p(2) - p(-2)]/(2 - (-2)) = (-2 - 10)/4 = -12/4 = -3. Choice A is correct because it properly calculates [p(2) - p(-2)]/(2 - (-2)) = -12/4 = -3, getting both the arithmetic and the sign right! Choice B has the magnitude right but the wrong sign. When p(2) is less than p(-2), the numerator is negative, giving a negative rate, not positive. The key formula for average rate of change is (y₂ - y₁)/(x₂ - x₁), which you might recognize as the slope formula! To use it: (1) identify your two points or endpoints, (2) subtract the y-values (later minus earlier), (3) subtract the x-values (later minus earlier), (4) divide. Keep the order consistent and you'll get the right answer every time!
A function T(n) gives the total number of minutes it takes to download n files. In this context, n must be a whole number (you can download 0 files, 1 file, 2 files, etc.). Which set best represents the domain of T?
Explanation: This question tests your understanding of what functions are, and how to determine their domains (possible inputs) and ranges (possible outputs). The domain is the set of all possible input values (x-values) that make sense for the function: for formulas, we exclude values that would cause division by zero or square roots of negatives, and in real-world contexts, we only include values that are realistic (like you can't have -3 people or 2.5 items if they're discrete). Since n represents the number of files downloaded, it must be a whole number—you can't download 2.5 files or -3 files! The domain includes 0 (downloading no files), 1, 2, 3, and so on. Choice C is correct because {0,1,2,3,...} represents exactly the non-negative integers (whole numbers), which are the only sensible values for counting files. Choice B includes negative integers like -1, -2, which don't make sense for file counts. When dealing with counting problems, your domain is usually the non-negative integers {0,1,2,3,...}. This set has a special name: the whole numbers or natural numbers including zero!
The graph shows the function f(x)=x+1x−2, which models the efficiency ratio of a machine as a function of operating speed x. Based on the graph and the mechanical context, what domain restriction is most appropriate?
Explanation: In mechanical contexts, operating speed must be positive (x>0). Since −1<0, the mathematical restriction x=−1 is automatically satisfied when we require x>0, making it redundant to state both conditions. Choice A allows negative speeds. Choice B is mathematically correct but unnecessarily redundant. Choice D incorrectly assumes negative efficiency ratios are meaningless - machines can operate inefficiently (negative ratios) and still function.
Perform the indicated operation and simplify: 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 (ba⋅dc=bdac). But here's the smart way: factor first, cancel common factors, THEN multiply—it keeps the numbers smaller and the result already simplified! For (x3⋅x+2x2), multiply to get (x(x+2)3x2), then cancel one x from numerator and denominator to leave (x+23x). Choice A correctly multiplies and simplifies to (x+23x) by canceling the common x factor. For example, choice B doesn't cancel the x, so it's not fully simplified—look for those common factors! 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.
An investment account loses 15% of its value every 6 months. Which expression correctly models the account balance after t years, and how should this be classified?
Explanation: Losing 15% means retaining 85%, so the factor is 0.85. Since this happens every 6 months, it occurs twice per year, giving A(t)=A0(0.85)2t. The annual multiplier is (0.85)2=0.7225, representing a 27.75% annual loss. Choice B has incorrect time scaling. Choice C uses 0.15 instead of 0.85. Choice D incorrectly suggests growth.
Rewrite the rational exponent expression x3/4 using radicals.
Explanation: This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Converting between radicals and rational exponents follows a simple pattern: the denominator of the exponent tells you which root (square root is 1/2, cube root is 1/3), and the numerator tells you what power. So ∛(x²) = x^(2/3) because we're taking the cube root (denominator 3) of x squared (numerator 2). For x^(3/4), the denominator 4 tells us we need a fourth root, and the numerator 3 tells us the power is 3. This gives us ⁴√(x³), which means the fourth root of x cubed. Choice C is correct because x^(3/4) means the fourth root (denominator 4) of x cubed (numerator 3), written as ⁴√(x³). Choice A incorrectly uses a cube root instead of a fourth root, while choice B uses a square root with the wrong power. The key to converting: denominator of exponent = which root, numerator = which power. So x^(3/4) means fourth root of x cubed: ⁴√(x³). To remember which is which, think '3 on top means power of 3, 4 on bottom means 4th root.' The fraction tells you everything!
A system consists of the equations y=x2−4x+3 and y=2x−5. If the system has two solutions, what is the sum of the x-coordinates of these solutions?
Explanation: Setting the equations equal: x2−4x+3=2x−5. Rearranging: x2−6x+8=0. Using Vieta's formulas, the sum of the roots equals the negative coefficient of x divided by the leading coefficient: −(−6)/1=6. Choice B results from forgetting the negative sign in Vieta's formula. Choice C comes from incorrectly setting up the quadratic as x2−2x+8=0. Choice D results from using 6/(−1) instead of (−(−6))/1.
A function is described as follows: It starts at the point (0,2), increases to a maximum at (3,5), then decreases and crosses the x-axis at (6,0). Where does the function change from increasing to decreasing?
Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. A maximum is the highest point on a graph (or on some portion of it), giving the largest y-value, while a minimum is the lowest point, giving the smallest y-value. For a parabola that opens up, the vertex is the minimum; if it opens down, the vertex is the maximum. In real-world problems, these tell you the best or worst outcome! To sketch a function with starting at (0,2), maximum at (3,5), x-intercept at (6,0), we: (1) Plot key points like intercepts and extrema: (0,2), (3,5), (6,0), (2) Note behavior: increasing on (0,3), decreasing on (3,6), (3) Connect with appropriate curve type, (4) Verify sketch shows all required features. The sketch doesn't need to be perfect, just show the main features clearly! Choice D correctly identifies the change from increasing to decreasing at x=3 because that's the maximum point where the behavior switches. Choice B confuses with the x-intercept: x=6 is where it crosses x-axis, but the switch happens at the peak—look for where the description says 'increases to a maximum' then 'decreases'! For sketching from verbal descriptions: (1) Make a checklist of all required features, (2) Plot any specific points given (intercepts, extrema), (3) Identify regions where function increases, decreases, is positive, is negative, (4) Connect the dots with the right curve type (line, parabola, etc.) making sure all features are visible. Don't worry about making it perfect—as long as the key features are clearly shown, you're good!
For the quadratic function h(x)=x2, which interval has the larger average rate of change: [0,2] or [2,4]?
Explanation: This question tests your understanding of average rate of change, which is a super important concept connecting slope, functions, and real-world rates like speed or growth. But for nonlinear functions like quadratics, the average rate of change can be different over different intervals. Over interval [0,2], the rate is [h(2) - h(0)]/(2 - 0) = (4 - 0)/2 = 2. Over interval [2,4], the rate is [h(4) - h(2)]/(4 - 2) = (16 - 4)/2 = 6. Comparing these: 6 > 2, so interval [2,4] has the larger average rate of change. This makes sense because the function is steeper there. Choice C is correct because it properly calculates and compares the rates to identify [2,4] as larger, getting the interpretation right! Choice A states the interpretation backwards: [0,2] has the smaller rate, not larger. Quick sanity check: if the function is increasing (going up) on your interval, the average rate should be positive. If decreasing (going down), it should be negative. If horizontal, it should be zero. Does your answer match what you see happening in the function? If not, recheck your work!
A streaming service charges \12permonthplusaone−timesign−upfeeof$5.ThetotalcostaftermmonthsisS(m) = 12m + 5.WhatisS(2)$?
Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values in a real-world context. In real-world contexts, a function like S(m)=12m+5 gives us a formula to calculate one quantity (like cost) from another (like number of months), and evaluating S(2) tells us the specific cost after 2 months. In this problem, S(2)=12(2)+5=24+5=29 dollars, which means the total cost after 2 months is 29 dollars. The function helps us quickly answer 'what if' questions by just plugging in different input values! Choice A is correct because it properly substitutes 2 for m in the function and calculates accurately: 24+5=29. Nice work if you got this! Choice B is a common slip-up: it substitutes correctly but then makes an arithmetic error, calculating 12(2) as 24 but forgetting to add the 5. Double-checking your arithmetic is always a good idea! In word problems, always state what your answer means: don't just write 'S(2)=29'—say 'S(2)=29 dollars, which is the total cost after 2 months.' This shows you understand what the math represents!
A gym charges a one-time sign-up fee of 30andthen18 per month. You have at most $150 to spend total. Write an inequality for the number of months you can afford, and find the maximum whole number of months.
Let m = the number of months.
Explanation: This question tests your ability to translate a real-world situation into a mathematical equation or inequality, solve it, and interpret the result in the original context. For inequalities, words like 'at most,' 'maximum,' 'no more than' signal ≤ (less than or equal), while 'at least,' 'minimum,' 'no less than' signal ≥ (greater than or equal). 'More than' means > (strict), and 'less than' means <. These key phrases tell you which inequality symbol to use! The context 'A gym charges a one-time sign-up fee of 30andthen18 per month. You have at most 150tospendtotal′usesthephrase′atmost150,' which signals ≤. Setting up: 30 + 18m ≤ 150. Solving: subtract 30 → 18m ≤ 120, divide by 18 → m ≤ 6.666. This means maximum of 6 whole months, since you can't have a fraction of a month here. Choice B is correct because it properly sets up the inequality from context, solves correctly, and interprets appropriately, giving maximum m=6 whole months. Choice A has the inequality symbol backwards or ignores the fee: perhaps doing 18m ≤ 150 → m ≤ 8.33 (to 8, but wrong), but 'at most 150′means≤afteradding30. These phrases can be tricky—memorizing which direction they go really helps! For inequalities, make a quick reference card: 'at most/maximum/no more than' → ≤ (can equal or be less), 'at least/minimum/no less than' → ≥ (can equal or be more), 'more than/over' → > (strictly greater), 'less than/under' → < (strictly less). Having these memorized means you'll never use the wrong symbol!
The volume of a spherical balloon is given by V(t)=34πr03⋅23t where r0 is the initial radius and t is time in hours. Which equivalent expression reveals how the radius changes with time?
Explanation: When you encounter exponential expressions involving geometric formulas, look for ways to factor out the geometric structure to reveal how individual dimensions change over time. The volume formula for a sphere is V=34πr3, where r is the radius. To find how the radius changes, you need to rewrite the given expression to match this standard form and identify what the radius equals at time t. Starting with V(t)=34πr03⋅23t, you can rewrite the exponential term: 23t=(2t)3. This gives you V(t)=34πr03⋅(2t)3. Now factor the cube: V(t)=34π(r0⋅2t)3. This matches the standard sphere volume formula where the radius is r0⋅2t, meaning the radius doubles every hour. Choice A correctly simplifies 23t=8t but doesn't reveal the radius pattern—it keeps the volume in terms of the initial radius cubed rather than showing the current radius. Choice B incorrectly states the radius grows linearly, but r0⋅2t represents exponential growth, not linear. Choice C introduces an incorrect factor of 3t that doesn't appear in the original expression and creates a nonsensical mixed growth pattern. Study tip: When working with exponential functions involving geometric formulas, use exponent rules like (ab)n=anbn and amn=(am)n to factor expressions and reveal how individual dimensions change over time.
On a coordinate plane, graph the quadratic function y=(x+1)2+2. What is the minimum value of the function?
Explanation: This question tests your understanding of how to identify key features of quadratic functions and their graphs, specifically the vertex and its relation to minimum value. 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 = 1(x + 1)² + 2, where we can read the vertex directly: it's (h, k) = (-1, 2); since a > 0, the minimum value is the y-coordinate of the vertex, which is 2—the turning point is the very bottom of the parabola! Choice C is correct because it properly identifies the minimum value as 2 by reading the k value from vertex form, since the parabola opens up. Great job identifying this! Choice A confuses the h value with the minimum—an easy mix-up! The minimum is the y-coordinate k=2, not the x-shift of -1. 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!
Write both the explicit formula and the recursive formula for the arithmetic sequence 4, 9, 14, 19, ….
Explanation: This question tests your understanding of arithmetic and geometric sequences and how to write them both recursively (each term from the previous) and explicitly (any term directly from its position). Arithmetic sequences have a constant difference between consecutive terms: you add (or subtract) the same amount each time. The explicit formula aₙ = a₁ + (n - 1)d uses the first term a₁ and the common difference d to find any term directly, while the recursive form a₁ = [value], aₙ₊₁ = aₙ + d describes how to get each term from the one before. Looking at the sequence 4, 9, 14, 19, ..., we find the common difference by subtracting consecutive terms: 9 - 4 = 5, 14 - 9 = 5, so d = 5. With first term a₁ = 4, the explicit formula is aₙ = 4 + (n - 1)·5. The recursive form is: a₁ = 4, aₙ₊₁ = aₙ + 5. Both formulas describe the pattern 'start at 4, add 5 each time'! Choice B is correct because it properly identifies the first term as a₁ = 4 and the common difference as d = 5, giving the explicit formula aₙ = 4 + 5(n-1) and recursive formula a₁ = 4, aₙ₊₁ = aₙ + 5. Great work! Choice A confuses arithmetic with geometric: this sequence adds the same amount each time, so it's arithmetic, not geometric. The pattern is 'add 5', not 'multiply by 5'. To identify arithmetic vs geometric: subtract consecutive terms (if differences are constant, it's arithmetic with d = that difference), or divide consecutive terms (if ratios are constant, it's geometric with r = that ratio). If neither pattern works, it's neither arithmetic nor geometric!
For sufficiently large x, which inequality will be true when comparing the linear function f(x)=100x and the exponential function h(x)=1.1x?
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 and g(x) = 1.1^x: showing calculations at several x-values. We find f(x) > g(x) for interval small x, but g(x) > f(x) for larger interval. The crossover occurs around x = approximate value where they equal, beyond which exponential dominates. Beyond this point, the exponential pulls away and never looks back—the gap between g(x) and f(x) grows without bound as x increases! Choice B correctly states that exponential exceeds linear for large x / shows understanding of growth hierarchy with specific correct reasoning or evidence. Choice D reverses the long-term hierarchy, saying linear grows faster than exponential long-term. This is backwards! For any exponential with base b > 1, no matter how large the linear coefficient, the exponential eventually exceeds it. Even 1000x < 1.001^x for large enough x (though the crossover happens at enormously large x). Common pitfall: don't assume the function that's largest at x = 1 or x = 5 will remain largest forever! Initial values can mislead. A polynomial might beat an exponential for small x, but EVENTUALLY (the key word!), the exponential always wins. Always check large x-values or think about the growth mechanism (additive vs multiplicative) to predict long-term behavior correctly.
Use completing the square to analyze u(x)=x2+10x+9. Which statement gives the correct vertex and axis of symmetry?
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. To complete the square for u(x) = x² + 10x + 9: half of 10 is 5, squared is 25. Adding and subtracting: u(x) = (x² + 10x + 25) - 25 + 9 = (x + 5)² - 16. The vertex form shows vertex at (-5, -16), and the axis of symmetry is the vertical line x = -5. Choice B correctly completes the square to get (x + 5)² - 16 showing vertex at (-5, -16) and axis at x = -5. Choice D makes an error completing the square: it calculates -25 + 9 as 16 instead of -16. When you add and subtract to complete the square, be careful with the arithmetic: -25 + 9 = -16, not 16! Completing the square reminder: for x² + 10x, the perfect square you add is (10/2)² = 5² = 25—half the middle coefficient, then square it. Then carefully compute the constant term: original constant (9) minus what you added (25) gives 9 - 25 = -16. Watch signs carefully!