Algebra 2 Quiz: Interpreting Sketching Key Features Of Functions

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Interpreting Sketching Key Features Of FunctionsQuestion 1 of 20

Sketch a graph of a function $y=f(x)$ that satisfies all of the following features:

$x$ -intercepts at $(-2,0)$ and $(4,0)$
$y$ -intercept at $(0,3)$
A relative maximum at $(1,5)$
End behavior: as $x\to\infty$ , $f(x)\to-\infty$ and as $x\to-\infty$ , $f(x)\to\infty$

Which option shows a sketch that matches these features?

A curve crossing the

x

-axis at

x=-2

and

x=4

, passing through

(0,3)

, peaking near

(1,5)

, rising on the left and falling on the right

A curve crossing the

x

-axis at

x=-2

and

x=4

, passing through

(0,3)

, with a relative minimum near

(1,5)

, falling on the left and rising on the right

A curve with

x

-intercepts at

(-2,0)

and

(4,0)

y

-intercept at

(3,0)

, and both ends falling

A curve crossing the

x

-axis at

x=-2

and

x=4

, passing through

(0,3)

, peaking near

(1,5)

, and rising on both ends

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Algebra 2 Quiz

Algebra 2 Quiz: Interpreting Sketching Key Features Of Functions

Practice Interpreting Sketching Key Features Of Functions in Algebra 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Interpreting Sketching Key Features Of Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Sketch a graph of a function $y=f(x)$ that satisfies all of the following features:

$x$ -intercepts at $(-2,0)$ and $(4,0)$
$y$ -intercept at $(0,3)$
A relative maximum at $(1,5)$
End behavior: as $x\to\infty$ , $f(x)\to-\infty$ and as $x\to-\infty$ , $f(x)\to\infty$

Which option shows a sketch that matches these features?

A curve crossing the $x$ -axis at $x=-2$ and $x=4$ , passing through $(0,3)$ , peaking near $(1,5)$ , rising on the left and falling on the right (correct answer)
A curve crossing the $x$ -axis at $x=-2$ and $x=4$ , passing through $(0,3)$ , with a relative minimum near $(1,5)$ , falling on the left and rising on the right
A curve with $x$ -intercepts at $(-2,0)$ and $(4,0)$ , $y$ -intercept at $(3,0)$ , and both ends falling
A curve crossing the $x$ -axis at $x=-2$ and $x=4$ , passing through $(0,3)$ , peaking near $(1,5)$ , and rising on both ends

Explanation: This question tests your ability to identify and interpret key features of functions—like intercepts, where they increase or decrease, maximum and minimum values, end behavior, and (for some functions) periodicity—from graphs, tables, or formulas. Key features tell the complete story of a function: intercepts show where it crosses the axes (starting value or zeros), increasing/decreasing intervals show where it's rising or falling, extrema show peaks and valleys (best/worst outcomes), and end behavior describes long-term trends. Each feature is described using x-values (intervals) or points (intercepts, extrema), never y-values alone for intervals—this is crucial! When we say 'increasing on (2, 5),' we mean 'for x-values from 2 to 5, the function is rising.' To match the features, the sketch must cross x at -2 and 4, y at (0,3), have a peak at (1,5), and show end behavior rising to the left (to ∞ as x→-∞) and falling to the right (to -∞ as x→∞). Choice A correctly shows a sketch with these features, including the peaking and appropriate end behavior directions. A distractor like choice B might swap the extremum to a minimum or reverse the end behavior, but verify each listed feature matches exactly. Sketching from features: (1) Make a checklist of all required features, (2) Plot any specific points given (intercepts, extrema), (3) Mark intervals where function increases, decreases, is positive, negative, (4) Note end behavior (where does it go as x → ±∞?), (5) Connect with smooth curve of appropriate type (line for linear, parabola for quadratic, etc.) showing all features. Your sketch doesn't need to be perfect—as long as all key features are clearly visible and correct, you've succeeded!

Question 2

A model for the value of a used laptop (in dollars) after $t$ years is $V(t)=900(0.8)^t.$ Describe the end behavior of $V(t)$ as $t \to \infty$ and interpret it in context.

As $t\to \infty$ , $V(t)\to -\infty$ ; the laptop's value becomes negative.
As $t\to \infty$ , $V(t)\to 900$ ; the laptop's value levels off at $900$ dollars.
As $t\to \infty$ , $V(t)\to 0$ ; the laptop's value approaches $0$ dollars over time. (correct answer)
As $t\to \infty$ , $V(t)\to \infty$ ; the laptop becomes more valuable over time.

V(t) = 900(0.8)^t

, as

t \to \infty

, since

0.8 < 1

(0.8)^t \to 0

, so

V(t) \to 0

, meaning the laptop's value approaches

0

dollars over time. Choice B correctly identifies the end behavior as

V(t) \to 0

t \to \infty

and interprets it as the value approaching

0

. Choice A incorrectly says it goes to

\infty

, perhaps confusing with growth (base

>1

). The interval confusion fix: intervals ALWAYS describe x-values (inputs), never y-values! 'Increasing on [2, 5]' means 'as x goes from 2 to 5, y is rising'—it describes the horizontal extent where behavior occurs. Similarly, 'positive on (-3, 4)' means 'for x between -3 and 4, the function is above the x-axis (y > 0).' If you catch yourself using y-values for intervals, stop and switch to x-values. This is one of the most common errors in working with key features!

Question 3

The temperature deviation from average (in °C) during a day is modeled by $T(t)=3\cos\left(\frac{\pi}{12}t\right)$ , where $t$ is hours after midnight. What is the period of $T(t)$ , and what does it mean in context?

Period $=\pi/12$ hours; the pattern repeats every $\pi/12$ hours
Period $=12$ hours; the pattern repeats every 12 hours
Period $=6$ hours; the pattern repeats every 6 hours
Period $=24$ hours; the pattern repeats every 24 hours (correct answer)

Explanation: This question tests your ability to identify and interpret key features of functions—like intercepts, where they increase or decrease, maximum and minimum values, end behavior, and (for some functions) periodicity—from graphs, tables, or formulas. Periodic functions repeat their pattern at regular intervals: if a function has period p, then f(x + p) = f(x) for all x. Sine and cosine are the classic examples with period 2π. To find values, use the period: if f has period 4 and f(1) = 3, then f(5) = f(1 + 4) = f(1) = 3, and f(9) = f(1 + 8) = f(1) = 3. The pattern repeats every p units, so you can 'wrap around' by adding or subtracting multiples of p! For T(t)=3 cos(π/12 t), the period is 2π divided by the coefficient of t, which is 2π/(π/12)=24 hours, meaning the temperature deviation pattern repeats every 24 hours, like a daily cycle. Choice B correctly identifies the period as 24 hours and interprets it as the pattern repeating every 24 hours in the context of a day. A mistake like in choice A might halve the period by miscounting the coefficient, but remember to divide 2π by the full angular speed π/12. The interval confusion fix: intervals ALWAYS describe x-values (inputs), never y-values! 'Increasing on [2, 5]' means 'as x goes from 2 to 5, y is rising'—it describes the horizontal extent where behavior occurs. Similarly, 'positive on (-3, 4)' means 'for x between -3 and 4, the function is above the x-axis (y > 0).' If you catch yourself using y-values for intervals, stop and switch to x-values. This is one of the most common errors in working with key features!

Question 4

A polynomial models the net revenue (in thousands of dollars) from ticket sales for an event: $R(x)=(x+1)(x-4)(x-6),$ where $x$ is the ticket price (in dollars). On what interval is $R(x)$ positive?

$(-\infty,-1)\cup(4,6)$
$( -1,4)\cup(4,6)$
$(0,\infty)$
$(-1,4)\cup(6,\infty)$ (correct answer)

Explanation: This question tests your ability to identify and interpret key features of functions—like intercepts, where they increase or decrease, maximum and minimum values, end behavior, and (for some functions) periodicity—from graphs, tables, or formulas. Key features tell the complete story of a function: intercepts show where it crosses the axes (starting value or zeros), increasing/decreasing intervals show where it's rising or falling, extrema show peaks and valleys (best/worst outcomes), and end behavior describes long-term trends. Each feature is described using x-values (intervals) or points (intercepts, extrema), never y-values alone for intervals—this is crucial! When we say 'increasing on (2, 5),' we mean 'for x-values from 2 to 5, the function is rising.' For R(x) = (x+1)(x-4)(x-6), roots are at x=-1,4,6; testing intervals shows positive on (-1,4) and (6,∞) since the leading coefficient is positive and sign changes at each root. Choice B correctly uses intervals with x-values to identify where R(x) > 0 as (-1,4) ∪ (6,∞). Choice A swaps the positive and negative intervals, perhaps from miscounting sign changes. The interval confusion fix: intervals ALWAYS describe x-values (inputs), never y-values! 'Increasing on [2, 5]' means 'as x goes from 2 to 5, y is rising'—it describes the horizontal extent where behavior occurs. Similarly, 'positive on (-3, 4)' means 'for x between -3 and 4, the function is above the x-axis (y > 0).' If you catch yourself using y-values for intervals, stop and switch to x-values. This is one of the most common errors in working with key features!

Question 5

A fireworks shell's height above the ground (in meters) $t$ seconds after launch is modeled by $h(t)=-5(t-2)^2+20$ , for $0\le t\le 5$ . Identify the maximum height of the shell and when it occurs.

Maximum height $20$ meters at $t=2$ seconds (correct answer)
Maximum height $-20$ meters at $t=2$ seconds
Maximum height $20$ meters at $t=-2$ seconds
Maximum height $15$ meters at $t=2$ seconds

Explanation: This question tests your ability to identify and interpret key features of functions—like intercepts, where they increase or decrease, maximum and minimum values, end behavior, and (for some functions) periodicity—from graphs, tables, or formulas. Key features tell the complete story of a function: intercepts show where it crosses the axes (starting value or zeros), increasing/decreasing intervals show where it's rising or falling, extrema show peaks and valleys (best/worst outcomes), and end behavior describes long-term trends. Each feature is described using x-values (intervals) or points (intercepts, extrema), never y-values alone for intervals—this is crucial! When we say 'increasing on (2, 5),' we mean 'for x-values from 2 to 5, the function is rising.' The function h(t) = -5(t-2)^2 + 20 is a downward-opening parabola with vertex form, so the maximum occurs at the vertex t=2, where h(2)=20, representing the peak height within the domain 0≤t≤5. Choice A correctly identifies the maximum height of 20 meters at t=2 seconds. A common mistake, like in choice B, is misinterpreting the negative coefficient as making the maximum negative, but the +20 shifts it positive. The interval confusion fix: intervals ALWAYS describe x-values (inputs), never y-values! 'Increasing on [2, 5]' means 'as x goes from 2 to 5, y is rising'—it describes the horizontal extent where behavior occurs. Similarly, 'positive on (-3, 4)' means 'for x between -3 and 4, the function is above the x-axis (y > 0).' If you catch yourself using y-values for intervals, stop and switch to x-values. This is one of the most common errors in working with key features!

Question 6

A roller coaster's height above ground (in meters) is modeled on the interval $0\le x\le 6$ by the function $H(x)=-x^2+6x+1$ , where $x$ is horizontal distance (in tens of meters). What is the absolute maximum of $H$ on $[0,6]$ (give the point)?

$(6,1)$
$(3,10)$ (correct answer)
10
$(0,1)$

Question 7

A function $F(x)$ models the net gain (in dollars) from selling $x$ tickets to an event. The graph of $F$ crosses the $x$ -axis at $x=50$ and $x=200$ . Between these values, the graph is above the $x$ -axis. Interpret the $x$ -intercepts in context.

At 50 and 200 tickets, the net gain is $0$ dollars (break-even points) (correct answer)
At 50 and 200 tickets, the net gain is maximized
At 50 and 200 tickets, the ticket price is $0$ dollars
At 50 and 200 tickets, the net gain is negative

Explanation: This question tests your ability to identify and interpret key features of functions—like intercepts, where they increase or decrease, maximum and minimum values, end behavior, and (for some functions) periodicity—from graphs, tables, or formulas. Key features tell the complete story of a function: intercepts show where it crosses the axes (starting value or zeros), increasing/decreasing intervals show where it's rising or falling, extrema show peaks and valleys (best/worst outcomes), and end behavior describes long-term trends. Each feature is described using x-values (intervals) or points (intercepts, extrema), never y-values alone for intervals—this is crucial! When we say 'increasing on (2, 5),' we mean 'for x-values from 2 to 5, the function is rising.' The x-intercepts at x=50 and x=200 mean F(50)=0 and F(200)=0, and since the graph is above between them, these are break-even points where net gain is zero. Choice A correctly interprets the x-intercepts in context as at 50 and 200 tickets, the net gain is $0 (break-even points). A common confusion like in choice B is mistaking intercepts for maxima, but intercepts are where y=0, not peaks—check if the graph crosses or touches the axis. The interval confusion fix: intervals ALWAYS describe x-values (inputs), never y-values! 'Increasing on [2, 5]' means 'as x goes from 2 to 5, y is rising'—it describes the horizontal extent where behavior occurs. Similarly, 'positive on (-3, 4)' means 'for x between -3 and 4, the function is above the x-axis (y > 0).' If you catch yourself using y-values for intervals, stop and switch to x-values. This is one of the most common errors in working with key features!

Question 8

A company's profit function is $P(x) = -2x^3 + 24x^2 - 72x + 50$ , where $x$ represents the number of items produced (in hundreds) and $P(x)$ is profit in thousands of dollars. Analyzing the end behavior and critical points, which statement best describes the company's long-term production strategy?

Production should be maximized because profit increases without bound as production increases indefinitely.
Production should avoid the range of 200-600 items where the function experiences negative growth rates.
Production should target exactly 200 items to achieve the global maximum profit of the function.
Production should be limited to avoid the inevitable decrease in profit at high production levels due to the negative leading coefficient. (correct answer)

Explanation: When analyzing polynomial functions in business contexts, you need to examine both the end behavior (determined by the leading term) and critical points to understand the complete picture of how the function behaves. For this profit function

P(x) = -2x^3 + 24x^2 - 72x + 50

, the leading term is

-2x^3

. Since the coefficient is negative and the degree is odd, as

x

approaches positive infinity,

P(x)

approaches negative infinity. This means that at extremely high production levels, the company will eventually experience massive losses due to factors like resource constraints, market saturation, or exponentially increasing costs. To find critical points, set

P'(x) = -6x^2 + 48x - 72 = 0

. Factoring gives

-6(x^2 - 8x + 12) = -6(x-2)(x-6) = 0

, so critical points occur at

x = 2

and

x = 6

(representing 200 and 600 items). Choice D correctly identifies that the negative leading coefficient creates inevitable profit decline at high production levels, making unlimited expansion dangerous. Choice A is wrong because profit doesn't increase without bound—the end behavior shows eventual losses. Choice B misinterprets the critical points; "negative growth rates" between 200-600 items doesn't mean this range should be avoided entirely. Choice C incorrectly claims 200 items gives the global maximum when you'd need to evaluate the function at both critical points to determine which yields higher profit. Strategy tip: For polynomial business functions, always check the leading coefficient first—it tells you the long-term sustainability of the strategy being modeled.

Question 9

A ball is thrown upward from a height of 6 feet. The function $h(t) = -16t^2 + 32t + 6$ models the height of the ball in feet after $t$ seconds. Which statement correctly interprets a key feature of this function in the context of the problem?

The ball reaches its maximum height of 22 feet at $t = 1$ second, after which it begins to decrease. (correct answer)
The ball reaches its maximum height of 32 feet at $t = 2$ seconds, representing the vertex of the parabola.
The ball hits the ground at $t = 6$ seconds, which corresponds to the $y$ -intercept of the function.
The ball maintains a constant upward velocity of 32 feet per second throughout its flight path.

Explanation: To find the maximum height, we complete the square or use the vertex formula. The vertex occurs at

t = -\frac{b}{2a} = -\frac{32}{2(-16)} = 1

. At

t = 1

h(1) = -16(1)^2 + 32(1) + 6 = 22

feet. This represents the maximum height since the parabola opens downward. Choice B incorrectly calculates both the time and height. Choice C confuses the

y

-intercept (initial height) with when the ball hits the ground. Choice D misinterprets the coefficient 32 as constant velocity rather than initial velocity.

Question 10

A car's fuel efficiency function is $E(s) = -0.02s^2 + 1.6s + 10$ where $E$ is miles per gallon and $s$ is speed in mph. The car operates between 20 mph and 80 mph. Which analysis correctly identifies the optimal operating conditions and efficiency behavior?

Maximum efficiency of 42 mpg occurs at 40 mph; efficiency decreases symmetrically on both sides of this speed within the operating range. (correct answer)
Maximum efficiency of 40 mpg occurs at 42 mph; efficiency is always increasing throughout the given operating range.
Maximum efficiency of 42 mpg occurs at 40 mph; efficiency is higher at 80 mph than at 20 mph due to the quadratic nature.
Maximum efficiency of 44 mpg occurs at 45 mph; the function achieves its theoretical maximum within the operating constraints.

Explanation: The vertex occurs at

s = -\frac{b}{2a} = -\frac{1.6}{2(-0.02)} = 40

mph. At

s = 40

E(40) = -0.02(40)^2 + 1.6(40) + 10 = -32 + 64 + 10 = 42

mpg. Since the parabola opens downward, this is the maximum. At the boundaries:

E(20) = -0.02(400) + 32 + 10 = 34

mpg and

E(80) = -0.02(6400) + 128 + 10 = 10

mpg. The function decreases on both sides of the vertex. Choice B has wrong values and incorrect behavior description. Choice C incorrectly compares endpoint values. Choice D has wrong maximum location and value.

Question 11

The temperature in a greenhouse follows the function $T(h) = 68 + 12\sin\left(\frac{\pi h}{12}\right)$ , where $T$ is the temperature in degrees Fahrenheit and $h$ is the number of hours after midnight. Based on this model, which feature correctly describes the temperature pattern?

The temperature oscillates between 56°F and 80°F with a period of 12 hours, reaching maximum at 6 AM.
The temperature oscillates between 56°F and 80°F with a period of 24 hours, reaching maximum at 6 AM. (correct answer)
The temperature oscillates between 68°F and 80°F with a period of 24 hours, reaching maximum at noon.
The temperature oscillates between 60°F and 76°F with a period of 12 hours, reaching maximum at midnight.

Explanation: The function has amplitude 12, so temperature ranges from

68-12=56°F

68+12=80°F

. The period is

\frac{2\pi}{\pi/12} = 24

hours. The maximum occurs when

\sin\left(\frac{\pi h}{12}\right) = 1

, which happens when

\frac{\pi h}{12} = \frac{\pi}{2}

, so

h = 6

(6 AM). Choice A has the wrong period. Choice C has the wrong temperature range and maximum time. Choice D has wrong temperature range, period, and maximum time.

Question 12

The function $g(x) = \log_2(x + 4) - 3$ represents the acidity level of a chemical solution where $x$ is the concentration of a compound in mg/L. Which combination of key features correctly describes the domain restrictions and asymptotic behavior in this context?

The function is defined for concentrations $x > -4$ mg/L with a vertical asymptote at $x = -4$ and horizontal asymptote at $y = -3$ .
The function is defined for concentrations $x \geq 0$ mg/L with a vertical asymptote at $x = 0$ and approaches $y = -3$ as $x$ increases.
The function is defined for concentrations $x > -4$ mg/L with a vertical asymptote at $x = -4$ and no horizontal asymptote since it increases without bound. (correct answer)
The function is defined for concentrations $x > 0$ mg/L with a vertical asymptote at $x = -4$ and approaches negative infinity as $x$ approaches zero.

Explanation: When analyzing logarithmic functions in real-world contexts, you need to consider both the mathematical domain and the practical constraints of the situation. For

g(x) = \log_2(x + 4) - 3

, start by identifying the domain and asymptotic behavior. The domain requires the argument of the logarithm to be positive:

x + 4 > 0

, so

x > -4

. This means the function is defined for concentrations greater than -4 mg/L. The vertical asymptote occurs where the argument equals zero, at

x = -4

. As

x

approaches -4 from the right,

g(x)

approaches negative infinity. As

x

increases without bound,

\log_2(x + 4)

also increases without bound, so

g(x)

has no horizontal asymptote—it increases indefinitely. Choice A incorrectly claims there's a horizontal asymptote at

y = -3

. The "-3" is just a vertical shift, not an asymptote. Choice B makes two errors: it restricts the domain to

x \geq 0

(ignoring that negative concentrations between -4 and 0 are mathematically valid) and again incorrectly identifies a horizontal asymptote. Choice D correctly identifies that practical concentrations should be positive, but wrongly states the vertical asymptote is at

x = -4

while claiming the domain is

x > 0

—these contradict each other. Choice C correctly identifies the domain as

x > -4

, the vertical asymptote at

x = -4

, and recognizes that logarithmic functions have no horizontal asymptote since they grow without bound. Study tip: For logarithmic functions

\log_b(x - h) + k

, the vertical asymptote is always at

x = h

, and logarithms never have horizontal asymptotes—they increase or decrease without bound.

Question 13

A function $f$ is described as follows: it starts at the point $(0,2)$ , increases until it reaches a relative maximum at $(2,6)$ , then decreases crossing the $x$ -axis at $(4,0)$ , continues decreasing to a relative minimum at $(6,-4)$ , and then increases thereafter. Which choice correctly gives the intervals where $f$ is increasing?

$(0,2)\cup(6,\infty)$ (correct answer)
$(0,2)\cup(4,6)$
$(2,6)$
$(-\infty,2)\cup(6,\infty)$

Explanation: This question tests your ability to identify and interpret key features of functions—like intercepts, where they increase or decrease, maximum and minimum values, end behavior, and (for some functions) periodicity—from graphs, tables, or formulas. Key features tell the complete story of a function: intercepts show where it crosses the axes (starting value or zeros), increasing/decreasing intervals show where it's rising or falling, extrema show peaks and valleys (best/worst outcomes), and end behavior describes long-term trends. Each feature is described using x-values (intervals) or points (intercepts, extrema), never y-values alone for intervals—this is crucial! When we say 'increasing on (2, 5),' we mean 'for x-values from 2 to 5, the function is rising.' From the description, the function increases from x=0 to the max at x=2, decreases to the min at x=6, then increases after x=6, so the increasing intervals are (0,2)∪(6,∞), noting it crosses at x=4 during decrease but that doesn't affect increasing intervals. Choice B correctly uses intervals with x-values to identify where f is increasing as (0,2)∪(6,∞). A distractor like choice A might include the decreasing interval after the max or confuse the crossing point, but increasing means where the slope is positive, separated by extrema. The interval confusion fix: intervals ALWAYS describe x-values (inputs), never y-values! 'Increasing on [2, 5]' means 'as x goes from 2 to 5, y is rising'—it describes the horizontal extent where behavior occurs. Similarly, 'positive on (-3, 4)' means 'for x between -3 and 4, the function is above the x-axis (y > 0).' If you catch yourself using y-values for intervals, stop and switch to x-values. This is one of the most common errors in working with key features!

Question 14

A sinusoidal model for ocean tide height (in meters) at a pier is $T(t)=1.5\cos\left(\frac{\pi}{6}t\right)+2,$ where $t$ is time in hours. What is the period of $T(t)$ , and what does it mean in context?

Period $=6$ hours; the tide pattern repeats every 6 hours.
Period $=\pi/6$ hours; the tide pattern repeats every $\pi/6$ hours.
Period $=2\pi$ hours; the tide pattern repeats every $2\pi$ hours.
Period $=12$ hours; the tide pattern repeats every 12 hours. (correct answer)

Explanation: This question tests your ability to identify and interpret key features of functions—like intercepts, where they increase or decrease, maximum and minimum values, end behavior, and (for some functions) periodicity—from graphs, tables, or formulas. Periodic functions repeat their pattern at regular intervals: if a function has period p, then f(x + p) = f(x) for all x. Sine and cosine are the classic examples with period 2π. To find values, use the period: if f has period 4 and f(1) = 3, then f(5) = f(1 + 4) = f(1) = 3, and f(9) = f(1 + 8) = f(1) = 3. The pattern repeats every p units, so you can 'wrap around' by adding or subtracting multiples of p! For T(t) = 1.5 cos(π/6 t) + 2, the period is 2π / (π/6) = 12 hours, meaning the tide height repeats every 12 hours. Choice B correctly identifies the period as 12 hours and interprets it as the tide pattern repeating every 12 hours. Choice A halves it to 6, perhaps confusing with the coefficient directly. The interval confusion fix: intervals ALWAYS describe x-values (inputs), never y-values! 'Increasing on [2, 5]' means 'as x goes from 2 to 5, y is rising'—it describes the horizontal extent where behavior occurs. Similarly, 'positive on (-3, 4)' means 'for x between -3 and 4, the function is above the x-axis (y > 0).' If you catch yourself using y-values for intervals, stop and switch to x-values. This is one of the most common errors in working with key features!

Question 15

Sketch a graph of a function $f(x)$ that models a hiker's elevation change (in hundreds of meters) versus time $x$ (in hours) and has all of these features:

$y$ -intercept at $(0,1)$
x-intercepts at $(2,0)$ and $(6,0)$
Increasing on $(0,4)$ and decreasing on $(4,8)$
A relative (and absolute) maximum at $(4,3)$
End behavior: as $x\to\infty$ , $f(x)\to-\infty$ and as $x\to-\infty$ , $f(x)\to-\infty$ Which sketch matches these features?

A downward-opening curve that starts at $(0,1)$ , crosses the x-axis at $x=2$ and $x=6$ , peaks at $(4,3)$ , and goes down on both ends. (correct answer)
An upward-opening curve that starts at $(0,1)$ , crosses the x-axis at $x=2$ and $x=6$ , has a minimum at $(4,3)$ , and goes up on both ends.
A curve that starts at $(1,0)$ , crosses the y-axis at $(1,0)$ , peaks at $(3,4)$ , and goes up on both ends.
A periodic wave with period 4 that crosses the x-axis at $x=2$ and $x=6$ and has repeating peaks.

Explanation: This question tests your ability to identify and interpret key features of functions—like intercepts, where they increase or decrease, maximum and minimum values, end behavior, and (for some functions) periodicity—from graphs, tables, or formulas. Key features tell the complete story of a function: intercepts show where it crosses the axes (starting value or zeros), increasing/decreasing intervals show where it's rising or falling, extrema show peaks and valleys (best/worst outcomes), and end behavior describes long-term trends. Let's verify each feature for the correct sketch: y-intercept (0,1) ✓, x-intercepts at (2,0) and (6,0) ✓, increasing on (0,4) means rising until x=4 ✓, decreasing on (4,8) means falling after x=4 ✓, maximum at (4,3) ✓, and both ends going to -∞ requires a downward-opening curve ✓. Choice A correctly describes a downward-opening parabola-like curve meeting all these requirements. Choice B incorrectly shows an upward-opening curve with wrong end behavior, while Choice C confuses intercept notation. Sketching from features: (1) Make a checklist of all required features, (2) Plot any specific points given (intercepts, extrema), (3) Mark intervals where function increases, decreases, (4) Note end behavior, (5) Connect with smooth curve showing all features!

Question 16

A toy car's vertical position (in centimeters) is modeled by $f(x)=x^4-4x^2$ , where $x$ is the horizontal position (in centimeters). Does $f$ have symmetry? What type?

Odd symmetry (symmetric about the origin)
No symmetry
Even symmetry (symmetric about the $y$ -axis) (correct answer)
Symmetric about the line $y=x$

Question 17

A company models its weekly profit (in thousands of dollars) by the function $P(x)=-x^2+6x-5$ , where $x$ is the number of weeks since a new product launch. Interpret the x-intercepts of $P$ in this context by identifying them as points.

The x-intercepts are $(1,0)$ and $(5,0)$ , meaning profit is $0$ at weeks 1 and 5. (correct answer)
The x-intercepts are $(0,1)$ and $(0,5)$ , meaning profit is $0$ at weeks 1 and 5.
The x-intercepts are $(1,0)$ and $(6,0)$ , meaning profit is $0$ at weeks 1 and 6.
The x-intercepts are $1$ and $5$ , meaning profit is $0$ at $P=1$ and $P=5$ .

Question 18

A function is described as follows: It starts at the point $(0,2)$ , increases to a relative maximum at $(3,5)$ , then decreases crossing the x-axis at $x=6$ , and continues decreasing as $x$ increases. As $x\to -\infty$ , the function decreases without bound. Sketch a graph showing all of these features.

A curve passing through $(0,2)$ , peaking at $(3,5)$ , crossing the x-axis at $(6,0)$ , rising to the right, and rising to the left
A curve passing through $(0,2)$ , peaking at $(3,5)$ , crossing the x-axis at $(6,0)$ , decreasing to the right, and decreasing to the left (correct answer)
A curve passing through $(2,0)$ , having a minimum at $(3,5)$ , crossing the x-axis at $(0,6)$ , and increasing to the right
A curve passing through $(0,2)$ , having a relative minimum at $(3,5)$ , crossing the x-axis at $(6,0)$ , and decreasing to the left but increasing to the right

Explanation: This question tests your ability to identify and interpret key features of functions—like intercepts, where they increase or decrease, maximum and minimum values, end behavior, and (for some functions) periodicity—from graphs, tables, or formulas. Key features tell the complete story of a function: intercepts show where it crosses the axes (starting value or zeros), increasing/decreasing intervals show where it's rising or falling, extrema show peaks and valleys (best/worst outcomes), and end behavior describes long-term trends. Each feature is described using x-values (intervals) or points (intercepts, extrema), never y-values alone for intervals—this is crucial! When we say 'increasing on (2, 5),' we mean 'for x-values from 2 to 5, the function is rising.' The description includes a y-intercept at (0,2), relative max at (3,5), x-intercept at (6,0), decreasing after x=3 and continuing as x increases (to -∞), and as x→-∞ decreasing to -∞, suggesting an overall shape falling on both ends. Choice B correctly captures all features: passing through (0,2), peaking at (3,5), crossing at (6,0), and decreasing to both left and right, matching the end behaviors to -∞. A distractor like choice A incorrectly states rising to the right and left, which would imply end behavior to +∞, contradicting the decreasing description. Sketching from features: (1) Make a checklist of all required features, (2) Plot any specific points given (intercepts, extrema), (3) Mark intervals where function increases, decreases, is positive, negative, (4) Note end behavior (where does it go as x → ±∞?), (5) Connect with smooth curve of appropriate type (line for linear, parabola for quadratic, etc.) showing all features. Your sketch doesn't need to be perfect—as long as all key features are clearly visible and correct, you've succeeded!

Question 19

A company's weekly profit (in thousands of dollars) is modeled by the function $P(x)=-x^2+6x-5$ , where $x$ is the number of online ads purchased that week. On what interval is the profit positive (i.e., $P(x)>0$ )?

$(1,5)$ (correct answer)
$(-\infty,1)\cup(5,\infty)$
$(0,1)\cup(5,\infty)$
$(0,1)\cup(0,5)$

Explanation: This question tests your ability to identify and interpret key features of functions—like intercepts, where they increase or decrease, maximum and minimum values, end behavior, and (for some functions) periodicity—from graphs, tables, or formulas. Key features tell the complete story of a function: intercepts show where it crosses the axes (starting value or zeros), increasing/decreasing intervals show where it's rising or falling, extrema show peaks and valleys (best/worst outcomes), and end behavior describes long-term trends. Each feature is described using x-values (intervals) or points (intercepts, extrema), never y-values alone for intervals—this is crucial! To find where profit is positive, we need to solve P(x) > 0, which means -x² + 6x - 5 > 0. Factoring: -(x² - 6x + 5) = -(x - 1)(x - 5) > 0, which means (x - 1)(x - 5) < 0. This inequality is satisfied when one factor is positive and the other negative, which occurs when 1 < x < 5. Choice A correctly identifies the interval (1,5) where the profit is positive—between 1 and 5 ads purchased, the company makes money. Choice B incorrectly suggests profit is positive when x < 1 or x > 5, but testing shows P(0) = -5 < 0 and P(6) = -5 < 0, so the profit is actually negative in those regions. The interval confusion fix: intervals ALWAYS describe x-values (inputs), never y-values! 'Positive on (1,5)' means 'for x between 1 and 5, the function is above the x-axis (y > 0).' If you catch yourself using y-values for intervals, stop and switch to x-values.

Question 20

A function $T(t)$ models the temperature difference (in °C) from a comfortable baseline in a greenhouse, where $t$ is hours after midnight. The temperature difference repeats every 24 hours. If $T(3)=2$ , what is $T(27)$ ?

$-2$
$2$ (correct answer)
$24$
$0$