Interpreting/Sketching Key Features of Functions - Algebra 1

The function $f(x)=x^2-4x-5$ models the height of a drone above a reference level (in meters) as a function of horizontal position $x$ (in meters). Identify the x-intercepts of $f(x)$ .
Answer: $(-1,0)$ and $(5,0)$ 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.
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?
Answer: $(0,15)$ ; the initial height of the ball at $t=0$ 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!
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?
Answer: $(-\infty,4)$ 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!
A company's profit function is $P(x) = -2x^2 + 40x - 150$ where $x$ is the number of items produced (in hundreds) and $P(x)$ is profit in thousands of dollars. For what range of production levels does the company earn a positive profit?
Answer: The company earns positive profit when producing between 500 and 1500 items 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.
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 $t\to\infty$ .
Answer: As $t\to\infty$ , $N(t)\to \infty$ 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|!
Based on this scenario, during which time interval is the distance function decreasing at the fastest rate?
Answer: The distance function is never decreasing during this 6-hour period based on the given information 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.
A roller coaster's height above the ground (in feet) is modeled by $H(x)=-2x^2+12x+5,$ where $x$ is the horizontal distance (in tens of feet) from the start. Identify the $x$ -intercepts of $H(x)$ .
Answer: $(0.5,0)$ and $(5.5,0)$ 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.
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 $H(x)$ ?
Answer: Maximum at $(2,9)$ 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.
A population model is $P(t)=500(1.2)^t$ , where $t$ is years since the start of the study. What does the $y$ -intercept represent in context?
Answer: The initial population at $t=0$ 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!
A function $f(x)$ is defined by the description: it has a minimum at $(2,-3)$ and is symmetric about the vertical line $x=2$ . Identify the axis of symmetry.
Answer: $x=2$ 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!
A company's weekly profit (in dollars) from selling $x$ items is modeled by $P(x)=-x^2+8x-12.$ On what interval is the profit function increasing?
Answer: $(-\infty,4)$ 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.
Does the function $f(x)=x^4-3x^2$ have symmetry? If so, what type?
Answer: Even (symmetric about the $y$ -axis) 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|!
The graph shown represents the temperature $T(h)$ in degrees Fahrenheit at different hours $h$ after midnight during a 24-hour period. Which statement correctly interprets the relative maximum that occurs around hour 14?
Answer: The temperature reaches a local high of 78¬∞F around 2:00 PM within this 24-hour period A relative maximum is a point where the function value is greater than nearby values, but not necessarily the global maximum. At hour 14 (2:00 PM), the temperature reaches 78¬∞F, which is higher than surrounding times, making it a relative maximum. The phrase 'local high' correctly describes a relative maximum. Choice A incorrectly assumes this is the absolute daily maximum. Choice B incorrectly suggests the temperature remains constant after the maximum. Choice C confuses the temperature value (78¬∞F) with the rate of change and misunderstands what happens at a maximum (rate of change is zero, not maximum).
A function $g(x)$ has the following properties: $g(0) = -3$ , $g(2) = 0$ , $g(4) = 0$ , and $g(6) = -3$ . The function is symmetric about the line $x = 3$ . Based on this information, which sketch best represents the key features that $g(x)$ must have?
Answer: A parabola opening downward with vertex at $(3, 1)$ and x-intercepts at $x = 2$ and $x = 4$ 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.
A ball is tossed upward from a platform. Its height above the ground (in meters) after $t$ seconds is modeled by the function $h(t)=-5t^2+20t+2$ for $t\ge 0$ . What does the maximum of $h(t)$ represent in this context?
Answer: The greatest height the ball reaches above the ground 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! In this context where the function models the height of a ball over time, the maximum of h(t) = -5t² + 20t + 2 represents the highest point the ball reaches during its flight. Since the coefficient of t² is negative (-5), the parabola opens downward, so the vertex is indeed a maximum. To find it, we use t = -b/(2a) = -20/(2×(-5)) = 2 seconds, giving h(2) = -5(4) + 20(2) + 2 = 22 meters. Choice B correctly interprets the maximum as 'The greatest height the ball reaches above the ground' because the maximum y-value (22 meters) represents the peak height during the ball's trajectory. Choice A confuses the y-intercept (initial height at t=0, which is 2 meters) with the maximum—the ball starts at 2 meters but goes much higher! 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!
A spring's displacement from equilibrium (in centimeters) is modeled by $s(t)=2\sin\left(\frac{\pi}{4}t\right)$ , where $t$ is time in seconds. On what interval from $t=0$ to $t=8$ is the displacement positive?
Answer: $(0,4)$ 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 positive/negative: identify where the graph is above/below the x-axis—those x-values form the interval. For this graph: with period 8, sine is positive from t=0+ to t=4 (excluding endpoints where zero). Choice A correctly identifies the positive interval as (0,4) because that's where sin(π/4 t) >0 in [0,8]. Choice B describes the wrong interval: it says positive on (0,2)∪(4,6), but checking where the graph is above the x-axis, we see it's actually on (0,4) continuously—always verify by checking specific points in your interval! 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.
A function is described as follows: It starts at the point $(0,2)$ , increases to a maximum at $(3,5)$ , then decreases and crosses the $x$ -axis at $(6,0)$ . Where does the function change from increasing to decreasing?
Answer: $x=3$ 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! To sketch a function with starting at (0,2), maximum at (3,5), x-intercept at (6,0), we: (1) Plot key points like intercepts and extrema: (0,2), (3,5), (6,0), (2) Note behavior: increasing on (0,3), decreasing on (3,6), (3) Connect with appropriate curve type, (4) Verify sketch shows all required features. The sketch doesn't need to be perfect, just show the main features clearly! Choice D correctly identifies the change from increasing to decreasing at x=3 because that's the maximum point where the behavior switches. Choice B confuses with the x-intercept: x=6 is where it crosses x-axis, but the switch happens at the peak—look for where the description says 'increases to a maximum' then 'decreases'! For sketching from verbal descriptions: (1) Make a checklist of all required features, (2) Plot any specific points given (intercepts, extrema), (3) Identify regions where function increases, decreases, is positive, is negative, (4) Connect the dots with the right curve type (line, parabola, etc.) making sure all features are visible. Don't worry about making it perfect—as long as the key features are clearly shown, you're good!
A piecewise function models the elevation (in meters) of a hiker along a trail: $E(x)=\begin{cases} 2x & \text{for } 0\le x\le 4,\\ - x+12 & \text{for } 4< x\le 10. \end{cases}$ On what interval is the elevation decreasing?
Answer: $(4,10]$ 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!
The figure shows the graph of $y = f(x)$ . Which statement about the intervals where $f(x)$ is positive and the intervals where $f(x)$ is increasing is most accurate?
Answer: The function is positive on $(-1, 3)$ and increasing on $(-\infty, 1) \cup (4, \infty)$ From the graph: $f(x) > 0$ (above x-axis) between the x-intercepts at $x = -1$ and $x = 3$ , so positive on $(-1, 3)$ . The function is increasing (sloping upward) from the left until the local maximum at $x = 1$ , then decreasing until the local minimum at $x = 4$ , then increasing again to the right. So $f(x)$ is increasing on $(-\infty, 1) \cup (4, \infty)$ . Choice B incorrectly identifies the increasing intervals. Choice C incorrectly identifies where the function is positive. Choice D incorrectly describes the increasing intervals.
Describe the end behavior of $f(x)=-2x^3+5x^2$ .
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 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. Looking at the graph: end behavior dominated by -2x³ term, so as x→∞, f(x)→-∞; as x→-∞, f(x)→∞. Choice B correctly describes the end behavior as x→∞, f(x)→-∞ and x→-∞, f(x)→∞ because for negative leading coefficient on odd degree, right end down, left end up. Choice A reverses the directions: it's the opposite for negative odd degree—always check the sign and parity of the leading term! 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|!

Start this quiz