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. For T(t) = 2cos(πt/6) + 5, the period is found from the coefficient of t inside cosine: period = 2π/(π/6) = 2π × 6/π = 12 hours. Choice B correctly identifies the period as 12 hours, meaning the tide pattern repeats every 12 hours. Choice A incorrectly gives 6 (perhaps from the denominator alone), while Choices C and D give periods in terms of π without proper calculation. The pattern repeats every p units, so you can 'wrap around' by adding or subtracting multiples of p! For trig functions, period = 2π/b where b is the coefficient of the variable.

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, evaluate s(-x): $s(-x)=(-x)^4$ $-3(-x)^2$ = $x^4$ $-3x^2$ = s(x), confirming even symmetry about the y-axis, meaning the position pattern mirrors across the starting line. Choice B correctly identifies even symmetry (symmetric about the y-axis). A common error like in choice A is assuming odd symmetry due to the position context, but only functions where s(-x)=-s(x) are odd, and this has even powers only. 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. 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.

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!

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 $V(t) = 900(0.8)^t$, 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 $V(t) \to 0$ as $t \to \infty$ 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!

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. 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.' 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!

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!

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!