12 multiple-choice questions, progressively harder.
The graph of y=g(x)y = g(x)y=g(x) is y=f(x)y = f(x)y=f(x) shifted right 444 and down 222. Written with fff, what is g(x)g(x)g(x)?
Solution
Correct answer: B
Right 444 replaces xxx with x−4x - 4x−4; down 222 subtracts 222 outside the function.
g(x)=f(x−4)−2g(x) = f(x - 4) - 2g(x)=f(x−4)−2
For y=(x+6)2−1y = (x + 6)^2 - 1y=(x+6)2−1, what is the vertex and how many x-intercepts does it have?
Correct answer: C
From x+6x + 6x+6 and −1-1−1, the vertex is (−6,−1)(-6, -1)(−6,−1). The parabola opens upward with lowest height −1<0-1 < 0−1<0, so it dips below the axis and crosses it twice.
lowest y=−1<0 ⇒ two x-intercepts\text{lowest } y = -1 < 0 \;\Rightarrow\; \text{two x-intercepts}lowest y=−1<0⇒two x-intercepts
Given g(x)=f(x−3)−5g(x) = f(x - 3) - 5g(x)=f(x−3)−5 and g(8)=0g(8) = 0g(8)=0, find f(5)f(5)f(5).
Correct answer: D
With g(x)=f(x−3)−5g(x) = f(x - 3) - 5g(x)=f(x−3)−5, at x=8x = 8x=8 the inside is 8−3=58 - 3 = 58−3=5.
g(8)=f(5)−5=0 ⇒ f(5)=5g(8) = f(5) - 5 = 0 \;\Rightarrow\; f(5) = 5g(8)=f(5)−5=0⇒f(5)=5
The x-intercepts of y=f(x)y = f(x)y=f(x) are (−2,0)(-2, 0)(−2,0) and (5,0)(5, 0)(5,0). Find the x-intercepts of y=f(x−4)y = f(x - 4)y=f(x−4).
The inside x−4x - 4x−4 is a right shift by 444, which moves both x-intercepts right by 444.
(−2+4, 0)=(2,0),(5+4, 0)=(9,0)(-2 + 4,\; 0) = (2, 0), \qquad (5 + 4,\; 0) = (9, 0)(−2+4,0)=(2,0),(5+4,0)=(9,0)
A translation sends (x,y)(x, y)(x,y) to (x+5,y−3)(x + 5, y - 3)(x+5,y−3). Which equation gives the new graph in terms of fff?
Correct answer: A
Points moving to (x+5,y−3)(x + 5, y - 3)(x+5,y−3) shift the graph right 555 and down 333. Right 555 makes the inside x−5x - 5x−5; down 333 subtracts 333 outside.
y=f(x−5)−3y = f(x - 5) - 3y=f(x−5)−3
The graph of g(x)=f(x+1)+6g(x) = f(x + 1) + 6g(x)=f(x+1)+6 passes through (2,10)(2, 10)(2,10). Which point lies on the graph of fff?
With g(x)=f(x+1)+6g(x) = f(x + 1) + 6g(x)=f(x+1)+6, at x=2x = 2x=2 the inside is 2+1=32 + 1 = 32+1=3.
g(2)=f(3)+6=10 ⇒ f(3)=4g(2) = f(3) + 6 = 10 \;\Rightarrow\; f(3) = 4g(2)=f(3)+6=10⇒f(3)=4
So (3,4)(3, 4)(3,4) lies on fff.
The graph of y=f(x)y = f(x)y=f(x) has a maximum at (−1,8)(-1, 8)(−1,8). After shifting to y=f(x−2)−5y = f(x - 2) - 5y=f(x−2)−5, where is the maximum?
Right 222 and down 555 move the maximum along with the graph.
(−1+2, 8−5)=(1,3)(-1 + 2,\; 8 - 5) = (1, 3)(−1+2,8−5)=(1,3)
The graph of y=f(x)y = f(x)y=f(x) passes through (a,b)(a, b)(a,b). After shifting left 333 and up 444, the image point is (1,9)(1, 9)(1,9). Find (a,b)(a, b)(a,b).
The image is left 333 and up 444 of the original, so undo it: add 333 to the input and subtract 444 from the output.
(1+3, 9−4)=(4,5)(1 + 3,\; 9 - 4) = (4, 5)(1+3,9−4)=(4,5)
The parabola y=(x−1)2+2y = (x - 1)^2 + 2y=(x−1)2+2 is shifted left 444 and down 666. Write the new equation.
Left 444 replaces xxx with x+4x + 4x+4, turning x−1x - 1x−1 into x+3x + 3x+3; down 666 subtracts 666 from the constant 222.
y=(x+3)2+(2−6)=(x+3)2−4y = (x + 3)^2 + (2 - 6) = (x + 3)^2 - 4y=(x+3)2+(2−6)=(x+3)2−4
A function fff satisfies f(0)=3f(0) = 3f(0)=3 and f(2)=7f(2) = 7f(2)=7. For g(x)=f(x−2)g(x) = f(x - 2)g(x)=f(x−2), what is g(2)g(2)g(2)?
With g(x)=f(x−2)g(x) = f(x - 2)g(x)=f(x−2), at x=2x = 2x=2 the inside is 2−2=02 - 2 = 02−2=0.
g(2)=f(0)=3g(2) = f(0) = 3g(2)=f(0)=3
A shift moves the point (5,−2)(5, -2)(5,−2) to (5,4)(5, 4)(5,4) with no horizontal movement. Which equation gives the new graph?
The input stays at 555 and the output rises from −2-2−2 to 444, a purely vertical shift.
4−(−2)=6 ⇒ y=f(x)+64 - (-2) = 6 \;\Rightarrow\; y = f(x) + 64−(−2)=6⇒y=f(x)+6
The graph of y=f(x)y = f(x)y=f(x) has a zero (x-intercept) at (6,0)(6, 0)(6,0). The graph is shifted left 222. Is the zero still a zero, and where?
A horizontal shift keeps every point at the same height, so a point on the x-axis stays on the x-axis. Left 222 moves it to input 6−2=46 - 2 = 46−2=4.
(6−2, 0)=(4,0)(6 - 2,\; 0) = (4, 0)(6−2,0)=(4,0)
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