Interpreting/Sketching Key Features of Functions - Algebra 2

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?
Answer: $(-1,4)\cup(6,\infty)$ 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!
A population model is given by $f(x)=-2x^3+5x$ , where $x$ is time in years. Describe the end behavior of $f$ .
Answer: As $x\to\infty$ , $f(x)\to-\infty$ and as $x\to-\infty$ , $f(x)\to\infty$ 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. For the cubic function f(x) = -2x³ + 5x, end behavior is determined by the leading term -2x³. As x → ∞, the term -2x³ dominates and becomes very negative (since we're multiplying a large positive x³ by -2), so f(x) → -∞. As x → -∞, we have a large negative value cubed (which gives a large negative) times -2 (which makes it positive), so f(x) → ∞. Choice B correctly describes this end behavior: as x → ∞, f(x) → -∞ and as x → -∞, f(x) → ∞. Choice A reverses these directions, while Choices C and D incorrectly suggest the function approaches 0 or both ends go to ∞. 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.
Consider the function $g(x)=x^4-3x^2.$ Does $g$ have symmetry? What type?
Answer: Yes; $g$ is even (y-axis symmetry). 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 check symmetry, compute g(-x) = $(-x)^4$ - 3 $(-x)^2$ = $x^4$ - $3x^2$ = g(x), so it's even with y-axis symmetry. Choice C correctly identifies g as even with y-axis symmetry. Choice A mistakes it for odd, which would require g(-x) = -g(x). 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!
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?
Answer: 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. 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!
A ball is thrown upward from a platform. Its height (in meters) after $t$ seconds is $h(t)=-5(t-2)^2+20.$ On what interval is h(t) increasing?
Answer: $(-\infty,2)$ 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 at t=2, so it increases for t < 2 (as t increases toward 2, h rises) and decreases for t > 2. Choice B correctly uses intervals with x-values to identify the increasing interval as $(-\infty, 2)$ . Choice C mistakenly uses y-values $(0,20)$ instead of t-values, which confuses the output with the input domain. 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!
A company models the number of active users (in thousands) by $U(x)=x^3-4x$ , where $x$ is years since launch. Does $U$ have symmetry? What type?
Answer: Odd symmetry (symmetric about the origin) 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 check symmetry, compute U(-x) = $(-x)^3$ - 4(-x) = - $x^3$ + 4x = -( $x^3$ - 4x) = -U(x), confirming odd symmetry about the origin. Choice B correctly identifies odd symmetry, meaning the graph looks the same after a 180-degree rotation around the origin, which fits this cubic function factored as x( $x^2$ - 4). A distractor like choice A suggests even symmetry, but that would require U(-x) = U(x), which doesn't hold here since the odd powers dominate. 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!
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)?
Answer: $(3,10)$ 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. For H(x) = -x² + 6x + 1, we complete the square to get H(x) = -(x-3)² + 10, showing the vertex is at (3,10), and since the parabola opens downward, this is the maximum on any interval containing x = 3. Choice A correctly identifies the absolute maximum as the point (3,10), meaning the roller coaster reaches its highest point of 10 meters at horizontal distance 30 meters (since x is in tens of meters). Choice D gives only the y-value without the x-coordinate, while Choice C confuses the maximum location. Sketching from features: when asked for extrema, always give the complete point (x,y) unless specifically asked for just the value—context matters!
A function is defined by $f(x)=x^4-3x^2$ . Does $f$ have symmetry? What type?
Answer: Even symmetry (y-axis symmetry) 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. To check for symmetry in f(x) = x⁴ - 3x², we test if f(-x) = f(x) (even symmetry) or f(-x) = -f(x) (odd symmetry). Computing f(-x) = (-x)⁴ - 3(-x)² = x⁴ - 3x² = f(x), which confirms even symmetry. This means the graph is symmetric about the y-axis—if you fold the graph along the y-axis, the two halves match perfectly. Choice B correctly identifies this as even symmetry (y-axis symmetry). Choice A would require f(-x) = -f(x), but we found f(-x) = f(x), so it's not odd. Choice D suggests periodicity, but polynomial functions are never periodic. The interval confusion fix: intervals ALWAYS describe x-values (inputs), never y-values! When checking symmetry, we're examining how the function behaves for opposite x-values: even functions have f(-x) = f(x), meaning the output is the same for x and -x.
A theme park models the number of visitors (in hundreds) $V(t)$ at time $t$ hours after opening as a periodic function. The pattern repeats every 6 hours, and $V(1)=8$ . What is $V(13)$ ?
Answer: 8 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. Since V has period 6 and we need V(13), we find how many complete periods fit: 13 = 1 + 2(6), so V(13) = V(1 + 12) = V(1) = 8. Choice A correctly identifies V(13) = 8 by using the periodicity property. Choice B incorrectly gives 2, perhaps confusing the number of periods with the value, while Choice C adds instead of using periodicity. The pattern repeats every p units, so you can 'wrap around' by adding or subtracting multiples of p! Think of it like a clock: after 12 hours, we're back where we started.
A population model is given by $P(x)=-2x^3+5x^2+1$ , where $x$ is years. Describe the end behavior of $P$ .
Answer: As $x\to\infty$ , $P(x)\to-\infty$ and as $x\to-\infty$ , $P(x)\to\infty$ 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 the cubic P(x) = - $2x^3$ + $5x^2$ + 1 with negative leading coefficient, as x→∞, the - $2x^3$ term dominates to -∞, and as x→-∞, it goes to +∞ (since $x^3$ → -∞, negative times negative is positive). Choice B correctly describes this end behavior: P(x)→-∞ as x→∞ and P(x)→∞ as x→-∞. A distractor like choice A reverses the directions, which would fit a positive leading coefficient instead. 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!
A function is shown on the coordinate plane as a parabola opening upward with vertex at $(2,-3)$ and passing through $(0,1)$ . Identify the $y$ -intercept of the function.
Answer: $(0,1)$ 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 y-intercept is the point where the graph crosses the y-axis, which is at x=0, and the description states it passes through (0,1), so that's the y-intercept. Choice B correctly identifies the y-intercept as (0,1). A mistake like in choice A might confuse it with an x-intercept or the vertex, but remember intercepts are specific axis crossings: y-intercept at x=0. 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!
The height of a tossed ball (in meters) after $t$ seconds is modeled by $h(t)=-5t^2+20t+1$ . Identify the y-intercept of $h$ and interpret it in context.
Answer: $(0,1)$ ; the ball starts at 1 meter when $t=0$ 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 find the y-intercept, substitute t=0 into h(t) = -5 $(0)^2$ + 20(0) + 1 = 1, so the point is (0,1), which means the ball's initial height is 1 meter at t=0 seconds. Choice A correctly identifies the y-intercept as (0,1) and interprets it as the starting height of the ball when t=0. A distractor like choice B confuses it with an x-intercept, which would be where h(t)=0 (when the ball hits the ground), but that's not asked here. 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!
A company's weekly profit (in thousands of dollars) is modeled by $P(x)=(x-1)(x-5)$ , where $x$ is the number of thousands of items sold that week. On what interval is the profit positive?
Answer: $(-\infty,1)\cup(5,\infty)$ 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 P(x)=(x-1)(x-5), a quadratic opening upward with roots at x=1 and x=5, it's positive outside the roots since the leading coefficient is positive, so on (-∞,1)∪(5,∞), meaning profit is positive when fewer than 1 thousand or more than 5 thousand items are sold. Choice B correctly uses intervals with x-values to identify where profit is positive as (-∞,1)∪(5,∞). A distractor like choice A might confuse the interval with the roots themselves, but positivity is determined by sign changes around the roots, not just between them. 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!
A ride's vertical position (in meters) is modeled by $y(t)=2\cos\left(\frac{\pi}{4}t\right)$ , where $t$ is time in seconds. On what interval(s) is $y(t)$ positive for $0\le t\le 8$ ?
Answer: $(0,2)\cup(6,8)$ 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 y(t) = 2 cos( (π/4) t ), the period is 8 seconds; over 0 to 8, y>0 where the cosine is positive, from t=0 to 2 (excluding 2) and t=6 to 8 (excluding 6), approximately (0,2) ∪ (6,8) since it equals zero at t=2 and 6. Choice B correctly identifies the intervals (0,2) ∪ (6,8) using x-values (time t) where the position is above zero. A distractor like choice C uses (0,4) ∪ (4,8), but includes intervals where cosine is negative, like t=2 to 4. 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!
A company models its daily profit (in dollars) by $P(x)=(x-2)(x+4)$ , where $x$ is the number of days since tracking began. On what interval is $P(x)$ negative?
Answer: $(-4,2)$ 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. From P(x) = (x-2)(x+4), the zeros are x = 2 and x = -4, and since this upward-opening parabola is negative between its zeros, P(x) < 0 on (-4,2). Choice B correctly identifies this interval using x-values, meaning the company has negative profit (loss) from day -4 to day 2. Choice C makes the critical error of describing this as 'y-values between -4 and 2,' while Choice A gives where P(x) > 0. The interval confusion fix: intervals ALWAYS describe x-values (inputs), never y-values! 'Negative on (-4,2)' means 'for x between -4 and 2, the function is below the x-axis (y < 0).'
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.
Answer: 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 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!
A skateboard ramp's cross-section is modeled by $f(x)=x^4-3x^2$ , where $x$ is horizontal distance (meters) from the center. Does $f$ have symmetry? What type?
Answer: Even symmetry (symmetric about the y-axis) 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 f(x) = $x^4$ - $3x^2$ , compute f(-x) = $(-x)^4$ - 3 $(-x)^2$ = $x^4$ - $3x^2$ = f(x), confirming even symmetry about the y-axis. Choice B correctly identifies even symmetry, meaning the graph is mirrored across the y-axis, which makes sense for a function with only even powers. A distractor like choice A suggests odd symmetry, but that requires f(-x) = -f(x), which fails here due to the even exponents. 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!
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?
Answer: Production should be limited to avoid the inevitable decrease in profit at high production levels due to the negative leading coefficient. 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.
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 $T(27)$ ?
Answer: $2$ 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. Since T has period 24 hours and T(3) = 2, we can find T(27) by recognizing that 27 = 3 + 24, which means T(27) = T(3 + 24) = T(3) = 2. The temperature difference at hour 27 is the same as at hour 3 because we've gone through exactly one complete 24-hour cycle. Choice B correctly identifies T(27) = 2. Choice A gives -2 (perhaps thinking the pattern inverts), Choice C gives 24 (confusing the period with the value), and Choice D gives 0. The pattern repeats every p units, so you can 'wrap around' by adding or subtracting multiples of p!
A ride-sharing company models driver earnings (in dollars) after $x$ hours by $E(x)=|x-5|-2$ . Identify the x-intercept(s) of $E$ (as points).
Answer: $(3,0)$ and $(7,0)$ 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. To find x-intercepts of E(x) = |x - 5| - 2, we set E(x) = 0 and solve: |x - 5| - 2 = 0, so |x - 5| = 2. This absolute value equation has two solutions: x - 5 = 2 (giving x = 7) or x - 5 = -2 (giving x = 3). The x-intercepts are the points (3, 0) and (7, 0). Choice A correctly identifies both x-intercepts as (3, 0) and (7, 0)—these are the points where the earnings function crosses the x-axis (zero earnings). Choice B incorrectly gives y-intercepts format, Choice C gives just x = 5 (the vertex's x-coordinate), and Choice D gives the vertex (5, -2) which is the minimum point, not an x-intercept. 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.

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