Algebra Quiz: Interpreting Sketching Key Features Of Functions

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.

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.

Question 1

1. Neither even nor odd
2. Even (symmetric about the $y$-axis) (correct answer)
3. Odd (symmetric about the origin)
4. Both even and odd

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Key features tell the story of a function: intercepts show where the function equals zero or starts, increasing/decreasing intervals show where it's rising or falling, maxima and minima show the peaks and valleys, and end behavior describes what happens as x gets very large or very small. Each feature reveals something important about the relationship being modeled. For g(x) = x⁴ - 2x², we need to check for symmetry. A function is even if g(-x) = g(x) for all x (symmetric about y-axis), and odd if g(-x) = -g(x) for all x (symmetric about origin). Let's check: g(-x) = (-x)⁴ - 2(-x)² = x⁴ - 2x² = g(x). Since g(-x) = g(x), the function is even! Choice A correctly identifies the function as 'Even (symmetric about the y-axis)' because replacing x with -x gives us the exact same function, which means the graph is a mirror image across the y-axis. Choice B would require g(-x) = -g(x), but we found g(-x) = g(x) instead—the function is definitely not odd! For symmetry checking: Even functions have only even powers of x (like x², x⁴) or constants. Odd functions have only odd powers (like x, x³). Mixed functions are usually neither. This function has only even powers (x⁴ and x²), so it's even!

Question 2

1. Maximum at $(2,-9)$
2. Maximum at $(2,9)$ (correct answer)
3. Maximum at $(-2,9)$
4. Maximum at $(9,2)$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. A maximum is the highest point on a graph (or on some portion of it), giving the largest y-value, while a minimum is the lowest point, giving the smallest y-value. For a parabola that opens up, the vertex is the minimum; if it opens down, the vertex is the maximum. In real-world problems, these tell you the best or worst outcome! For this downward-opening parabola H(x) = -(x-2)^2 + 9, the vertex at (2,9) is the maximum, meaning the highest point on the roller coaster track is 9 meters at x=2 meters horizontally. Choice A correctly identifies the maximum as (2,9) because the vertex form shows the peak at x=2, H(2)=9. Choice D identifies the maximum but at the wrong location: (-2,9) might come from misreading the vertex, but it's at x=2—make sure you're finding the right extreme! Quick reference for graph features: (1) Intercepts—where graph crosses axes, write as points (x, y), (2) Increasing/decreasing—read left to right, going up = increasing, going down = decreasing, state as x-intervals, (3) Positive/negative—above x-axis = positive, below = negative, state as x-intervals, (4) Maximum/minimum—highest/lowest points, give as points (x, y) or just y-value if asked, (5) End behavior—what happens at far left and far right of graph.

Question 3

1. As $x\to\infty$, $f(x)\to\infty$ and as $x\to-\infty$, $f(x)\to-\infty$
2. As $x\to\infty$, $f(x)\to-\infty$ and as $x\to-\infty$, $f(x)\to\infty$ (correct answer)
3. As $x\to\infty$, $f(x)\to 0$ and as $x\to-\infty$, $f(x)\to 0$
4. As $x\to\infty$, $f(x)\to\infty$ and as $x\to-\infty$, $f(x)\to\infty$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Key features tell the story of a function: intercepts show where the function equals zero or starts, increasing/decreasing intervals show where it's rising or falling, maxima and minima show the peaks and valleys, and end behavior describes what happens as x gets very large or very small. Each feature reveals something important about the relationship being modeled. For f(x) = -3x³ + 2x, the end behavior is determined by the leading term -3x³. Since this is a negative odd-degree term, as x → ∞ (x gets very large positive), -3x³ → -∞ (very large negative), and as x → -∞ (x gets very large negative), -3x³ → ∞ (very large positive). The +2x term doesn't affect end behavior because x³ grows much faster than x. Choice B correctly states 'As x→∞, f(x)→-∞ and as x→-∞, f(x)→∞' because the negative cubic function goes down on the right and up on the left. Choice A reverses this behavior—that would be true for a positive cubic like +3x³, not a negative one! End behavior shortcut for polynomials: look at the leading term (highest degree). Negative odd power (-x, -x³), left goes up, right goes down. This creates the characteristic 'S-shape' of cubic functions, just flipped when negative!

Question 4

1. $(4,\infty)$
2. $(-\infty,4)$ (correct answer)
3. $(-\infty,\infty)$
4. $(-\infty,0)$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. To identify where a function is increasing, look at the graph from left to right: if the graph is going upward (climbing), the function is increasing on that interval. If it's going downward (falling), it's decreasing. These intervals are described using the x-values, like 'increasing on (2, 5)' means as x goes from 2 to 5, the y-values are rising. For the profit function P(x) = -(x-4)² + 9, this is a parabola in vertex form with vertex at (4, 9). Since the coefficient of the squared term is negative (-1), the parabola opens downward. This means the function increases as we approach the vertex from the left and decreases as we move away from the vertex to the right. Therefore, the function is increasing for all x-values less than 4, which we write as the interval (-∞, 4). Choice B correctly identifies the increasing interval as (-∞, 4) because for a downward-opening parabola, the function rises from the left until it reaches its maximum at x = 4. Choice A gives (4, ∞), which is actually where the function is decreasing—after reaching the maximum at x = 4, the profit drops as we sell more items, perhaps due to oversupply or increased costs. It's easy to confuse which side is increasing! For parabolas, remember: if it opens down (negative coefficient), it increases on the left of the vertex and decreases on the right. If it opens up (positive coefficient), it decreases on the left and increases on the right. The vertex is always the turning point where the behavior changes!

Question 5

1. $(0,15)$; the initial height of the ball at $t=0$ (correct answer)
2. $(15,0)$; the time when the ball hits the ground
3. $(0,-15)$; the height below ground at $t=0$
4. $15$; the time when the ball is thrown

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Intercepts are special points: the y-intercept (0, b) is where the graph crosses the y-axis (this is the starting value when x = 0), and x-intercepts (a, 0) are where the graph crosses the x-axis (these are the zeros—where the function equals zero). In context, intercepts often have important meanings like 'initial value' or 'when does the quantity reach zero?' In this context where the function models the height of a ball thrown upward over time, the y-intercept at (0, 15) means the initial height of the ball is 15 meters when t=0, before any time has passed. Choice B correctly identifies the y-intercept as (0,15) and interprets it as the initial height of the ball at t=0 because plugging t=0 into h(t) gives h(0)=15, representing the starting point. Choice A confuses the y-intercept with an x-intercept: (15,0) would be where height is zero, like when the ball hits the ground, but that's not the y-intercept—always check by setting the input to zero for y-intercept! Context interpretation trick: intercepts often mean 'starting value' (y-intercept) or 'when does it reach zero' (x-intercept). Maxima/minima often mean 'best/worst case' or 'peak/valley.' Increasing means 'getting better' or 'growing,' decreasing means 'getting worse' or 'shrinking.' Translate the math features into the context language!

Question 6

1. $y=-3$
2. $x=-3$
3. $y=2$
4. $x=2$ (correct answer)

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Key features tell the story of a function: intercepts show where the function equals zero or starts, increasing/decreasing intervals show where it's rising or falling, maxima and minima show the peaks and valleys, and end behavior describes what happens as x gets very large or very small. Each feature reveals something important about the relationship being modeled. The function has a minimum at (2, -3) and is symmetric about the vertical line x = 2. For a parabola or any function with vertical line symmetry, the axis of symmetry is the vertical line that acts as a mirror for the graph. Since we're told the function is symmetric about x = 2, this is precisely the axis of symmetry! Choice C correctly identifies the axis of symmetry as x = 2 because this is the vertical line about which the function is symmetric—it's the line that would fold the graph onto itself. Choice A gives y = 2, which would be a horizontal line—but axes of symmetry for functions are typically vertical lines of the form x = a, not horizontal lines! For functions with a vertex (like parabolas), the axis of symmetry always passes through the vertex. Since the minimum is at (2, -3), the axis of symmetry must be the vertical line x = 2. This makes sense: the graph is the same distance from this line on both sides!

Question 7

1. As $t\to\infty$, $N(t)\to 200$
2. As $t\to\infty$, $N(t)\to 0$
3. As $t\to\infty$, $N(t)\to \infty$ (correct answer)
4. As $t\to\infty$, $N(t)\to -\infty$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Key features tell the story of a function: intercepts show where the function equals zero or starts, increasing/decreasing intervals show where it's rising or falling, maxima and minima show the peaks and valleys, and end behavior describes what happens as x gets very large or very small. Each feature reveals something important about the relationship being modeled. For end behavior of this exponential function N(t) = 200 · (1.5)^t, since the base 1.5 > 1, as t increases to infinity, N(t) grows without bound to infinity, modeling unlimited bacteria growth over time. Choice D correctly describes the end behavior as t → ∞, N(t) → ∞ because exponential growth with base >1 keeps multiplying and gets larger. Choice C misreads the growth: claiming N(t) → -∞ might confuse with decay, but since base >1, it's growth, not decay to negative—check the base! End behavior shortcut for polynomials: look at the leading term (highest degree). If it's positive even power (x², x⁴), both ends go up. Negative even power (-x², -x⁴), both ends go down. Positive odd power (x, x³), left goes down, right goes up. Negative odd power (-x, -x³), left goes up, right goes down. The leading term dominates for large |x|!

Question 8

1. $(4,\infty)$
2. $(-\infty,4)$ (correct answer)
3. $(-\infty,-4)$
4. $(0,\infty)$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. To identify where a function is increasing, look at the graph from left to right: if the graph is going upward (climbing), the function is increasing on that interval. If it's going downward (falling), it's decreasing. These intervals are described using the x-values, like 'increasing on (2, 5)' means as x goes from 2 to 5, the y-values are rising. For the profit function P(x) = -x² + 8x - 12, this is a downward-opening parabola (negative x² coefficient), which increases from the left up to its vertex, then decreases afterward. The vertex occurs at x = -b/(2a) = -8/(2(-1)) = 4, so the function increases on (-∞, 4) and decreases on (4, ∞). Choice B correctly identifies the increasing interval as (-∞, 4) because for a downward-opening parabola, the function rises from negative infinity up to the vertex at x = 4. Choice A gives (4, ∞), which is actually where the function is decreasing—after the peak at x = 4, the profit starts falling as we produce too many items. To avoid interval confusion: intervals ALWAYS use x-values (the inputs), never y-values! When we say 'increasing on (-∞, 4),' we mean 'as x goes from negative infinity to 4, y is rising.' For parabolas, remember: if it opens down, it increases then decreases; if it opens up, it decreases then increases.

Question 9

1. $(0,5)$ and $(6,5)$
2. $(0,-5)$ and $(6,-5)$
3. $(0.5,0)$ and $(5.5,0)$ (correct answer)
4. $(0,0.5)$ and $(0,5.5)$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Intercepts are special points: the y-intercept (0, b) is where the graph crosses the y-axis (this is the starting value when x = 0), and x-intercepts (a, 0) are where the graph crosses the x-axis (these are the zeros—where the function equals zero). In context, intercepts often have important meanings like 'initial value' or 'when does the quantity reach zero?' To find x-intercepts of H(x) = -2x² + 12x + 5, we set H(x) = 0 and solve: -2x² + 12x + 5 = 0. Using the quadratic formula: x = [-12 ± √(144 - 4(-2)(5))] / (2(-2)) = [-12 ± √(144 + 40)] / (-4) = [-12 ± √184] / (-4) = [-12 ± 2√46] / (-4) ≈ 0.5 and 5.5. So the x-intercepts are approximately (0.5, 0) and (5.5, 0), representing where the roller coaster is at ground level. Choice C correctly identifies the x-intercepts as (0.5, 0) and (5.5, 0) because these are the points where the height function equals zero—where the roller coaster touches the ground. Choice A gives points with y-coordinate 5, but x-intercepts must have y-coordinate 0 by definition—they're where the graph crosses the x-axis, not some other horizontal line. Context interpretation: x-intercepts in height problems often mean 'when does the object reach ground level?' Here, the coaster starts underground, rises above ground at x = 0.5, and returns to ground at x = 5.5.

Question 10

1. The maximum possible population
2. The growth factor per year
3. The time when the population reaches 0
4. The initial population at $t=0$ (correct answer)

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Intercepts are special points: the y-intercept (0, b) is where the graph crosses the y-axis (this is the starting value when x = 0), and x-intercepts (a, 0) are where the graph crosses the x-axis (these are the zeros—where the function equals zero). In context, intercepts often have important meanings like 'initial value' or 'when does the quantity reach zero?' In this context where the function models population growth, the y-intercept at (0,500) means the initial population at t=0 is 500. Choice B correctly interprets the y-intercept as the initial population at t=0 because plugging t=0 gives P(0)=500. Choice A confuses y-intercept with growth factor: the 1.2 is the growth factor, while 500 is the starting value—remember, y-intercept is f(0)! Context interpretation trick: intercepts often mean 'starting value' (y-intercept) or 'when does it reach zero' (x-intercept). Maxima/minima often mean 'best/worst case' or 'peak/valley.' Increasing means 'getting better' or 'growing,' decreasing means 'getting worse' or 'shrinking.' Translate the math features into the context language!

Question 11

1. Odd (symmetric about the origin)
2. Neither even nor odd
3. Even (symmetric about the $y$-axis) (correct answer)
4. Both even and odd

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Key features tell the story of a function: intercepts show where the function equals zero or starts, increasing/decreasing intervals show where it's rising or falling, maxima and minima show the peaks and valleys, and end behavior describes what happens as x gets very large or very small. Each feature reveals something important about the relationship being modeled. For symmetry, check if f(-x) = f(x) for even (y-axis symmetry) or f(-x) = -f(x) for odd (origin symmetry); here f(-x) = (-x)^4 - 3(-x)^2 = x^4 - 3x^2 = f(x), so it's even. Choice C correctly identifies it as even (symmetric about the y-axis) because replacing x with -x gives the same function. Choice A confuses even with odd: for odd, it should be f(-x) = -f(x), but here it's equal, not negative—always compute f(-x) to check! End behavior shortcut for polynomials: look at the leading term (highest degree). If it's positive even power (x², x⁴), both ends go up. Negative even power (-x², -x⁴), both ends go down. Positive odd power (x, x³), left goes down, right goes up. Negative odd power (-x, -x³), left goes up, right goes down. The leading term dominates for large |x|!

Question 12

1. At $x=-2$
2. At $x=2$
3. At $x=3$ (correct answer)
4. At $x=5$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. To identify where a function is increasing, look at the graph from left to right: if the graph is going upward (climbing), the function is increasing on that interval. If it's going downward (falling), it's decreasing. These intervals are described using the x-values, like 'increasing on (2, 5)' means as x goes from 2 to 5, the y-values are rising. For the absolute value function f(x) = |x - 3| - 2, the basic shape is a V with its vertex (turning point) at x = 3. To the left of x = 3, the function decreases (going down the left side of the V), and to the right of x = 3, it increases (going up the right side of the V). So the function changes from decreasing to increasing exactly at x = 3. Choice C correctly identifies x = 3 as where the function changes from decreasing to increasing because this is the vertex of the absolute value function—the point of the V where it stops going down and starts going up. Choice B suggests x = 2, but at x = 2 the function is still decreasing as we approach the vertex at x = 3—we haven't reached the turning point yet. Quick tip for absolute value functions: f(x) = |x - h| + k has its vertex (turning point) at x = h. The function decreases for x < h and increases for x > h. Here, h = 3, so that's our turning point!

Question 13

1. The time when the population reaches 0
2. The initial population at $t=0$ (correct answer)
3. The population after 1 hour
4. The time when the population doubles

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Intercepts are special points: the y-intercept (0, b) is where the graph crosses the y-axis (this is the starting value when x = 0), and x-intercepts (a, 0) are where the graph crosses the x-axis (these are the zeros—where the function equals zero). In context, intercepts often have important meanings like 'initial value' or 'when does the quantity reach zero?' For B(t) = 3·2^t, the y-intercept occurs when t = 0. Substituting: B(0) = 3·2⁰ = 3·1 = 3. Since B(t) represents population in thousands and t represents time in hours, the y-intercept of 3 means there were 3,000 bacteria at time t = 0—the initial population when observations began. Choice B correctly interprets the y-intercept as the initial population at t = 0 because the y-intercept always represents the function's value when the input variable equals zero, which in time-based contexts means the starting condition. Choice C might seem tempting since it mentions 'after 1 hour,' but that would be B(1) = 3·2¹ = 6, not the y-intercept—the y-intercept specifically requires t = 0, not t = 1! Context interpretation trick: intercepts often mean 'starting value' (y-intercept) or 'when does it reach zero' (x-intercept). In exponential growth/decay problems, the y-intercept is almost always the initial amount before any growth or decay has occurred. Always substitute x = 0 (or t = 0) to find the y-intercept!

Question 14

1. $[0,4]$
2. $(4,10]$ (correct answer)
3. $[0,10]$
4. $(0,4)$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. To identify where a function is increasing, look at the graph from left to right: if the graph is going upward (climbing), the function is increasing on that interval. If it's going downward (falling), it's decreasing. These intervals are described using the x-values, like 'increasing on (2, 5)' means as x goes from 2 to 5, the y-values are rising. For this piecewise function, let's analyze each piece: For 0 ≤ x ≤ 4, E(x) = 2x, which has positive slope 2, so it's increasing. For 4 < x ≤ 10, E(x) = -x + 12, which has negative slope -1, so it's decreasing. Therefore, the elevation increases from x = 0 to x = 4 (going uphill), then decreases from x = 4 to x = 10 (going downhill). Choice B correctly identifies (4, 10] as the interval where elevation is decreasing because on this portion of the trail, the function E(x) = -x + 12 has a negative slope, meaning the hiker is descending. Choice A gives [0, 4], but this is actually where the function is increasing—the hiker is climbing upward with E(x) = 2x on this interval. For piecewise functions: analyze each piece separately! Look at the formula for each interval and determine if it's increasing (positive slope for linear) or decreasing (negative slope). The behavior can change at the breakpoints!

Question 15

1. As $t$ approaches infinity, $P(t)$ approaches infinity, and the population triples each hour (correct answer)
2. As $t$ approaches infinity, $P(t)$ approaches 600, and the population increases by 200 each hour
3. As $t$ approaches infinity, $P(t)$ approaches infinity, and the population doubles each hour
4. As $t$ approaches negative infinity, $P(t)$ approaches 0, and the initial population was 200

Explanation: For the exponential function $P(t) = 200 \cdot 3^t$, as $t \to \infty$, $P(t) \to \infty$ since the base 3 > 1. The population triples every hour because $P(t+1) = 200 \cdot 3^{t+1} = 200 \cdot 3^t \cdot 3 = 3 \cdot P(t)$. Choice B is wrong because exponential functions don't approach horizontal asymptotes when the base > 1, and the growth is multiplicative, not additive. Choice C has correct end behavior but wrong growth factor. Choice D describes mathematically correct behavior for negative time but is irrelevant to the growth context.

Question 16

1. The company earns positive profit when producing more than 10 hundred items
2. The company earns positive profit when producing between 5 and 15 hundred items
3. The company earns positive profit when producing between 500 and 1500 items (correct answer)
4. The company earns positive profit when producing fewer than 20 hundred items

Explanation: When you encounter a profit function and need to find when profit is positive, you're solving a quadratic inequality. The key is finding where $P(x) > 0$, which means determining where the parabola is above the x-axis. To find when $P(x) = -2x^2 + 40x - 150 > 0$, first find where the profit equals zero by solving $-2x^2 + 40x - 150 = 0$. Dividing by -2 gives us $x^2 - 20x + 75 = 0$. This factors as $(x - 5)(x - 15) = 0$, so the roots are $x = 5$ and $x = 15$. Since the coefficient of $x^2$ is negative (-2), this parabola opens downward, meaning profit is positive between the roots. Therefore, profit is positive when $5 < x < 15$. Remember that $x$ represents hundreds of items, so this means between 500 and 1500 items, making C correct. Choice A suggests profit is positive only above 1000 items, missing the upper bound where profit becomes negative again. Choice B correctly identifies the range in hundreds but fails to convert back to actual items produced. Choice D claims profit is positive below 2000 items, ignoring that profit is also negative below 500 items. The key strategy here is remembering that quadratic inequalities with downward-opening parabolas are positive between their roots, and always check your units carefully. Many algebra problems involve converting between different scales (hundreds vs. actual quantities), so read the variable definitions closely.

Question 17

1. The balloon reaches its maximum height of 144 feet at $t = 2$ seconds after launch (correct answer)
2. The balloon hits the ground exactly 6 seconds after being launched from the platform
3. The balloon's height is decreasing most rapidly during the first second after launch
4. The balloon returns to its initial launch height after exactly 4 seconds of flight

Explanation: To find the maximum height, we complete the square or use the vertex formula. For $h(t) = -16t^2 + 64t + 80$, the vertex occurs at $t = -\frac{64}{2(-16)} = 2$. At $t = 2$: $h(2) = -16(4) + 64(2) + 80 = -64 + 128 + 80 = 144$ feet. Choice B is wrong because solving $-16t^2 + 64t + 80 = 0$ gives $t = 5$ (not 6). Choice C is wrong because the balloon is increasing in height during the first 2 seconds. Choice D is wrong because the initial height is 80 feet, and $h(4) = 80$, so it returns to launch height at $t = 4$, but this takes 4 seconds, not exactly 4 seconds as a special feature.

Question 18

1. A parabola opening upward with vertex at $(3, 1)$ and x-intercepts at $x = 2$ and $x = 4$
2. A parabola opening downward with vertex at $(3, 1)$ and x-intercepts at $x = 2$ and $x = 4$ (correct answer)
3. A parabola opening upward with vertex at $(3, -4)$ and x-intercepts at $x = 1$ and $x = 5$
4. A parabola opening downward with vertex at $(3, -4)$ and x-intercepts at $x = 1$ and $x = 5$

Explanation: Given the symmetry about $x = 3$ and the points provided: $g(0) = g(6) = -3$ (symmetric about $x = 3$), and $g(2) = g(4) = 0$ (x-intercepts). Since the function has value $-3$ at the symmetric points $x = 0$ and $x = 6$, and $0$ at $x = 2$ and $x = 4$, the vertex at $x = 3$ must be above the x-intercepts. The parabola opens downward with vertex at $(3, 1)$. Choice A has correct vertex and intercepts but wrong orientation. Choice C has wrong vertex location and intercepts. Choice D has wrong vertex and intercepts despite correct downward orientation.

Question 19

1. From hour 0 to hour 2, because this represents the longest continuous period of travel
2. From hour 3 to hour 5, because the cyclist travels the greatest total distance during this period
3. From hour 2 to hour 3, because the function remains constant at 15 miles throughout this interval
4. The distance function is never decreasing during this 6-hour period based on the given information (correct answer)

Explanation: Analyzing each interval: Hours 0-2: distance increases from 0 to 15 miles (increasing). Hours 2-3: distance stays at 15 miles (constant). Hours 3-5: distance increases from 15 to 25 miles (increasing). Hours 5-6: distance stays at 25 miles (constant). The distance from home never decreases because the cyclist never moves closer to home than their current position. Choice A describes an increasing interval. Choice B describes another increasing interval. Choice C describes a constant interval, not decreasing.

Question 20

1. $(-1,0)$ and $(5,0)$ (correct answer)
2. $(0,-5)$ and $(0,5)$
3. $(0,-1)$ and $(0,5)$
4. $(-5,0)$ and $(1,0)$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Intercepts are special points: the y-intercept (0, b) is where the graph crosses the y-axis (this is the starting value when x = 0), and x-intercepts (a, 0) are where the graph crosses the x-axis (these are the zeros—where the function equals zero). In context, intercepts often have important meanings like 'initial value' or 'when does the quantity reach zero?' To find the x-intercepts of f(x) = x² - 4x - 5, set f(x) = 0 and solve: factoring gives (x-5)(x+1) = 0, so x = 5 or x = -1, meaning points (-1,0) and (5,0); in the drone context, these represent positions where height is zero relative to the reference. Choice A correctly identifies the x-intercepts as (-1,0) and (5,0) because those are the points where the graph crosses the x-axis, found by solving f(x)=0. Choice B gives the y-values instead of the x-values for the interval: (0,-5) and (0,5) look like y-intercepts, but x-intercepts are where y=0, not x=0—remember, x-intercepts are points with y=0! Quick reference for graph features: (1) Intercepts—where graph crosses axes, write as points (x, y), (2) Increasing/decreasing—read left to right, going up = increasing, going down = decreasing, state as x-intervals, (3) Positive/negative—above x-axis = positive, below = negative, state as x-intervals, (4) Maximum/minimum—highest/lowest points, give as points (x, y) or just y-value if asked, (5) End behavior—what happens at far left and far right of graph.