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!

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!

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 p(x) = $-(x-5)^2$ + 16 is a downward-opening parabola with vertex at x=5, so it increases for x-values less than 5 (to the left of the vertex) and decreases for x greater than 5. Choice B correctly identifies the increasing interval as (-∞,5) using x-values, meaning profit rises as items sold increase up to 5. A common distractor like choice C uses y-values (0,16), which incorrectly describes output range instead of input intervals. 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!

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!

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!

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!

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!

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!

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!

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 g(x)=-2x^3 +5x, the end behavior is determined by the leading term -2x^3: as $x \to \infty$, $-2(\infty)^3 \to -\infty$; as $x \to -\infty$, $-2(-\infty)^3 \to -2(-\infty) \to \infty$, since $(-\infty)^3 = -\infty$ and negative times negative is positive. Choice B correctly describes the end behavior as $x \to \infty$, $g(x) \to -\infty$ and $x \to -\infty$, $g(x) \to \infty$. A distractor like choice A might flip the signs by ignoring the negative coefficient, but always check the leading term's sign and degree for long-term trends. 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!