Function is given by .
Function is shown on the coordinate plane as a parabola with x-intercepts at and .
Which statement correctly compares the x-intercepts of and ?
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Algebra · Learn by Concept
Review real example questions for Comparing Functions Represented In Different Ways in Algebra.
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Function f is given by f(x)=−x2+2x+3.
Function g is shown on the coordinate plane as a parabola with x-intercepts at x=−1 and x=3.
Which statement correctly compares the x-intercepts of f and g?
Function f is given by f(x)=−x2+2x+3.
Function g is shown on the coordinate plane as a parabola with x-intercepts at x=−1 and x=3.
Which statement correctly compares the x-intercepts of f and g?
Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. For function f given by f(x)=-x^2+2x+3, solve for x-intercepts by setting to zero: roots at x=-1 and x=3; for function g graphed with x-intercepts at -1 and 3, they match exactly. Choice C correctly identifies that both have x-intercepts at -1 and 3. If you picked choice B, that's understandable—check the quadratic formula or factoring to confirm f's roots. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Example: to compare y-intercepts, find where x = 0 in the formula, look where the graph crosses the y-axis, or find y when x = 0 in the table! Quick y-intercept trick: in a formula, set x = 0 and calculate. On a graph, see where it crosses the y-axis. In a table, find the y-value when x = 0. Three different methods, same property! Similarly, for comparing slopes of linear functions: read m from y = mx + b, calculate rise/run from a graph, or find Δy/Δx from consecutive table entries.
Function f is given algebraically by f(x)=2x−3. Function g is shown by the table below.
Table for g:
Which function has the larger y-intercept?
Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare y-intercepts, calculate f(0) = 2*0 - 3 = -3 from the formula for f, and read g(0) = 1 directly from the table for g, showing that 1 > -3. Choice B correctly identifies that function g has the larger y-intercept because 1 > -3. Don't worry if you mixed up the intercept with another point like (2,5)—just remember the y-intercept is always at x=0, so double-check that value in each representation. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values—for example, to compare y-intercepts, find where x=0 in the formula, look where the graph crosses the y-axis, or find y when x=0 in the table! Know what each representation shows best: formulas are great for calculating specific values and seeing patterns in the equation; graphs excel at showing maximums, minimums, and overall shape; tables are perfect for finding exact values at specific points; descriptions summarize key features—use each representation's strengths!
Function h is described verbally as: “a linear function with slope −4 and y-intercept 2.”
Function k is given by the table:
x: −1, 0, 1
k(x): 5, 3, 1
Which function has the steeper slope (greater slope magnitude)?
Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare slope magnitudes, note the verbal description gives h a slope of -4 (magnitude 4); for k's table, calculate slope as (3-5)/(0-(-1)) = -2 or (1-3)/(1-0) = -2 (magnitude 2), so 4 > 2. Choice A correctly identifies that function h has the steeper slope because |-4| > |-2|. If you thought k's slope was -1, that's okay—just practice calculating change in y over change in x from table points to get more comfortable. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Quick y-intercept trick: in a formula, set x=0 and calculate; on a graph, see where it crosses the y-axis; in a table, find the y-value when x=0—three different methods, same property! Similarly, for comparing slopes of linear functions: read m from y=mx+b, calculate rise/run from a graph, or find Δy/Δx from consecutive table entries.
Function f is described verbally as: “A parabola that opens upward with vertex at (2,−5).”
Function g is given by g(x)=(x−2)2−3.
Which function has the lower minimum value?
Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. For function f described as upward parabola with vertex (2,-5), minimum is -5; for function g given by g(x)=(x-2)^2-3, vertex at (2,-3) so minimum -3, compare -5 and -3. Choice B correctly identifies that function f has the lower minimum value because -5 < -3. If you chose choice A, remember lower means more negative, so -5 is below -3. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Example: to compare y-intercepts, find where x = 0 in the formula, look where the graph crosses the y-axis, or find y when x = 0 in the table! Know what each representation shows best: formulas are great for calculating specific values and seeing patterns in the equation; graphs excel at showing maximums, minimums, and overall shape; tables are perfect for finding exact values at specific points; descriptions summarize key features. Use each representation's strengths!
Function f is given by f(x)=x2−4x+1.
Function g is given in the table.
Which function has the smaller minimum value?
Table for g(x):
Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. For function f given by f(x)=x^2-4x+1, complete the square or use vertex formula x=-b/(2a)=2, then f(2)=-3 as minimum; for function g in the table, the lowest value is 1 at x=2, so compare -3 and 1. Choice A correctly identifies that function f has the smaller minimum value because -3 < 1. If you went with choice C, that's okay—just scan the table carefully for the smallest y-value. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Example: to compare y-intercepts, find where x = 0 in the formula, look where the graph crosses the y-axis, or find y when x = 0 in the table! Know what each representation shows best: formulas are great for calculating specific values and seeing patterns in the equation; graphs excel at showing maximums, minimums, and overall shape; tables are perfect for finding exact values at specific points; descriptions summarize key features. Use each representation's strengths!
Function p is given by p(x)=x2−6x+8.
Function q is described verbally as: “A parabola that opens upward with vertex at (1,−3).”
Which function has the smaller minimum value?
Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. For p(x) = x² - 6x + 8, complete the square: p(x) = (x-3)² - 9 + 8 = (x-3)² - 1, so the vertex is at (3, -1) and the minimum value is -1. Function q is described as having vertex at (1, -3), and since it opens upward, -3 is its minimum value. Since -3 < -1, function q has the smaller minimum. Choice B correctly identifies that function q has the smaller minimum value because -3 is less than -1. If you chose A, be careful with negative numbers—remember that -3 is smaller than -1 on the number line! Quick vertex trick: for ax² + bx + c, the x-coordinate of the vertex is -b/(2a), then substitute to find the y-coordinate. For verbal descriptions, the vertex coordinates are often given directly. Use each representation's strengths!
Function f is given by f(x)=3x+6.
Function g is shown by the table:
x: −2, −1, 0, 1
g(x): 4, 2, 0, −2
Which function has the larger x-intercept?
Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare x-intercepts, solve 3x + 6 = 0 for f to get x = -2; for g's table, note g(0) = 0, indicating x-intercept at 0 (and table suggests it's linear, crossing once), so 0 > -2. Choice B correctly identifies that function g has the larger x-intercept because 0 > -2. You might have thought f's intercept was 2 from mis-solving, but always set y=0 and solve for x carefully—you've got this! When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Similarly, for comparing slopes of linear functions: read m from y=mx+b, calculate rise/run from a graph, or find Δy/Δx from consecutive table entries.
Function f is given by f(x)=(x+1)2−4.
Function g is described verbally as: “A parabola that opens upward and has x-intercepts at x=−1 and x=3.”
Which function has more x-intercepts?
Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. For f(x) = (x+1)² - 4, expand to get f(x) = x² + 2x + 1 - 4 = x² + 2x - 3. Setting f(x) = 0: x² + 2x - 3 = 0, which factors as (x+3)(x-1) = 0, giving x-intercepts at x = -3 and x = 1 (two intercepts). Function g is described as having x-intercepts at x = -1 and x = 3 (also two intercepts). Both functions have exactly 2 x-intercepts! Choice C correctly identifies that they have the same number of x-intercepts (both have 2). If you chose differently, remember to count all x-intercepts—parabolas can have 0, 1, or 2 x-intercepts depending on how they're positioned! Know what each representation shows best: verbal descriptions often list x-intercepts directly, while for formulas you need to solve f(x) = 0. Quick tip: if a parabola is given in factored form like a(x-r)(x-s), the x-intercepts are at x = r and x = s!
Function f is described verbally as: “A linear function with slope −4 and y-intercept 2.”
Function g is given algebraically by g(x)=x+2.
Which function has the greater rate of change (slope)?
Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. For function f described verbally with slope -4, that's the rate of change; for function g given by g(x)=x+2, the slope is 1 from the coefficient of x, so we compare -4 and 1 to see which is greater. Choice B correctly identifies that function g has the greater rate of change because 1 > -4. If you chose choice A, that's understandable—negative slopes can be tricky, but greater means larger algebraically, so positive beats negative. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Example: to compare y-intercepts, find where x = 0 in the formula, look where the graph crosses the y-axis, or find y when x = 0 in the table! Quick y-intercept trick: in a formula, set x = 0 and calculate. On a graph, see where it crosses the y-axis. In a table, find the y-value when x = 0. Three different methods, same property! Similarly, for comparing slopes of linear functions: read m from y = mx + b, calculate rise/run from a graph, or find Δy/Δx from consecutive table entries.
Function h is described verbally as: “A linear function with slope −2 and y-intercept 5.”
Function k is given by k(x)=21x+1.
Which function has the greater slope?
Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. Function h is described verbally as having slope -2, while function k is given by k(x) = ½x + 1, which is in slope-intercept form y = mx + b where m = ½ is the slope. Comparing slopes: -2 versus ½. Choice B correctly identifies that function k has the greater slope because ½ > -2 (positive numbers are always greater than negative numbers). If you picked A, remember that negative numbers are always less than positive numbers, regardless of their absolute values. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (read from verbal description, identify m in y = mx + b form), (3) Compare the extracted values. Know what each representation shows best: verbal descriptions summarize key features directly, while formulas in slope-intercept form make the slope immediately visible as the coefficient of x!