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Algebra Help: Interpreting Sketching Key Features Of Functions

Review real example questions for Interpreting Sketching Key Features Of Functions in Algebra.

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A roller coaster's height (in meters) above the ground is modeled by $H(x)=-(x-2)^2+9$ , where $x$ is the horizontal distance (in meters) from the start of a section of track. What is the maximum of $H(x)$ ?

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Question 1

Maximum at $(2,-9)$
Maximum at $(2,9)$ (correct answer)
Maximum at $(-2,9)$
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 2

A ball is thrown upward from a platform. Its height above the ground after $t$ seconds is modeled by $h(t)=-5t^2+20t+15$ , where $h$ is in meters. What is the y-intercept of $h(t)$ , and what does it represent in context?

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

A population of bacteria (in thousands) after $t$ hours is modeled by $B(t)=3\cdot 2^t$ . What does the $y$ -intercept represent in this context?

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

Describe the end behavior of the function $f(x)=-3x^3+2x$ .

As $x\to\infty$ , $f(x)\to\infty$ and as $x\to-\infty$ , $f(x)\to-\infty$
As $x\to\infty$ , $f(x)\to-\infty$ and as $x\to-\infty$ , $f(x)\to\infty$ (correct answer)
As $x\to\infty$ , $f(x)\to 0$ and as $x\to-\infty$ , $f(x)\to 0$
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 5

A company's weekly profit (in thousands of dollars) depends on the number of ads $x$ it runs and is modeled by $P(x)=-2(x-3)^2+18$ , for $0\le x\le 6$ . On what interval is $P(x)$ increasing?

$(3,6)$
$[0,3]$ (correct answer)
$[0,6]$
$(0,3)$

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 quadratic function P(x) = -2(x-3)^2 + 18, which is a downward-opening parabola with vertex at x=3, the function increases to the left of the vertex (from x=0 to x=3) and decreases to the right (from x=3 to x=6), so the increasing interval within the domain is [0,3]. Choice B correctly identifies the increasing interval as [0,3] because before the vertex at x=3, as x increases, P(x) rises from P(0)=0 to P(3)=18. Choice C gives the y-values instead of the x-values for the interval: [0,6] might seem like the range of outputs, but when we say 'increasing on [0,3],' we mean 'for x-values from 0 to 3, the function increases.' The interval describes the input values (x), not the output values (y). This is a super common confusion! To avoid interval confusion: intervals ALWAYS use x-values (the inputs), never y-values! When we say 'increasing on [2, 5],' we mean 'as x goes from 2 to 5, y is rising.' Similarly, 'positive on (-3, 2)' means 'for x between -3 and 2, y > 0.' The interval describes the horizontal extent, not the vertical range.

Question 6

A piecewise function models the elevation (in meters) of a hiker along a trail:

2x & \text{for } 0\le x\le 4,\\ - x+12 & \text{for } 4< x\le 10. \end{cases}

On what interval is the elevation decreasing?

$[0,4]$
$(4,10]$ (correct answer)
$[0,10]$
$(0,4)$

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 7

A company's profit (in thousands of dollars) from selling $x$ hundred items is modeled by $P(x)=-x^2+6x-5$ . On what interval is the profit function increasing?

$(3,\infty)$
$(-\infty,3)$ (correct answer)
$(-\infty,0)$
$(-\infty,6)$

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. To find where the function is increasing/decreasing: looking at the graph from left to right, identify where it goes up/down—those x-values form the interval. For this graph: the quadratic opens downward with vertex at x=3, so increasing on (-∞,3) and decreasing on (3,∞). Choice B correctly identifies the increasing interval as (-∞,3) because before the vertex at x=3, the function rises as x increases. Choice A confuses increasing with decreasing: the function is actually decreasing on (3,∞) because as we move left to right after x=3, the graph goes down. It's easy to mix these up—always read the graph from left to right to determine which is which! To avoid interval confusion: intervals ALWAYS use x-values (the inputs), never y-values! When we say 'increasing on [2, 5],' we mean 'as x goes from 2 to 5, y is rising.' Similarly, 'positive on (-3, 2)' means 'for x between -3 and 2, y > 0.' The interval describes the horizontal extent, not the vertical range. 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 8

A company's weekly profit (in dollars) from selling $x$ items is modeled by $P(x)=-(x-4)^2+9$ . On what interval is the profit function increasing?

$(4,\infty)$
$(-\infty,4)$ (correct answer)
$(-\infty,\infty)$
$(-\infty,0)$

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 9

A bacteria culture's population is modeled by $N(t)=200\cdot(1.5)^t$ , where $t$ is time in hours. Describe the end behavior as $t\to\infty$ .

As $t\to\infty$ , $N(t)\to 200$
As $t\to\infty$ , $N(t)\to 0$
As $t\to\infty$ , $N(t)\to \infty$ (correct answer)
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 10

The profit (in dollars) from selling $x$ items is modeled by $P(x)=-2x^2+24x-40$ for $0 \leq x \leq 20$ . On what interval is $P(x)$ increasing?

$(0, 20)$
$(-\infty, 6)$
$(0, 6)$ (correct answer)
$(6, 20)$

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 $P(x) = -2x^2 + 24x - 40$ , we need to find where it's increasing. Since this is a downward-opening parabola (negative coefficient of $x^2$ ), it increases from the left boundary up to its vertex, then decreases. The vertex occurs at $x = -\frac{b}{2a} = -\frac{24}{2 \times (-2)} = 6$ . Since the domain is $0 \leq x \leq 20$ , the function increases from x = 0 to x = 6. Choice B correctly identifies the interval as (0, 6) because the profit rises as we go from selling 0 items to selling 6 items, after which it starts to decrease. Choice D gives (0, 20), which would mean the function increases over the entire domain—but this misses that after x = 6, the profit actually decreases as selling too many items reduces profit! To avoid interval confusion: intervals ALWAYS use x-values (the inputs), never y-values! When we say 'increasing on [0, 6],' we mean 'as x goes from 0 to 6, y is rising.' The interval describes the horizontal extent, not the vertical range.