12 multiple-choice questions, progressively harder.
The graph of y=f(x)y = f(x)y=f(x) is shifted right 222 and up 555. Which equation gives the new graph?
Solution
Correct answer: B
Right 222 replaces xxx with x−2x - 2x−2 (minus inside), and up 555 adds 555 outside.
y=f(x−2)+5y = f(x - 2) + 5y=f(x−2)+5
The point (5,−2)(5, -2)(5,−2) is on y=f(x)y = f(x)y=f(x). Which point is on y=f(x+1)−4y = f(x + 1) - 4y=f(x+1)−4?
Correct answer: C
The x+1x + 1x+1 inside is left 111 (subtract 111 from the input), and −4-4−4 outside is down 444.
(5−1, −2−4)=(4,−6)(5 - 1,\; -2 - 4) = (4, -6)(5−1,−2−4)=(4,−6)
What is the vertex of y=(x−4)2+3y = (x - 4)^2 + 3y=(x−4)2+3?
Correct answer: D
Vertex form y=(x−h)2+ky = (x - h)^2 + ky=(x−h)2+k has vertex (h,k)(h, k)(h,k). Here h=4h = 4h=4 and k=3k = 3k=3.
vertex=(4,3)\text{vertex} = (4, 3)vertex=(4,3)
What is the vertex of y=(x+2)2−5y = (x + 2)^2 - 5y=(x+2)2−5?
Read x+2x + 2x+2 as x−(−2)x - (-2)x−(−2), so h=−2h = -2h=−2, and the constant gives k=−5k = -5k=−5.
vertex=(−2,−5)\text{vertex} = (-2, -5)vertex=(−2,−5)
How is the graph of y=f(x+4)−2y = f(x + 4) - 2y=f(x+4)−2 related to the graph of y=f(x)y = f(x)y=f(x)?
Correct answer: A
The inside x+4x + 4x+4 shifts the graph left 444; the outside −2-2−2 shifts it down 222.
y=f(x+4)−2 ⇒ left 4, down 2y = f(x + 4) - 2 \;\Rightarrow\; \text{left } 4,\ \text{down } 2y=f(x+4)−2⇒left 4, down 2
A function fff has domain 1≤x≤51 \le x \le 51≤x≤5. What is the domain of y=f(x−2)y = f(x - 2)y=f(x−2)?
A right shift by 222 slides the whole domain right by 222, adding 222 to each end.
1≤x≤5 ⟶ 3≤x≤71 \le x \le 5 \;\longrightarrow\; 3 \le x \le 71≤x≤5⟶3≤x≤7
A function fff has range −2≤y≤4-2 \le y \le 4−2≤y≤4. What is the range of y=f(x)+5y = f(x) + 5y=f(x)+5?
A shift up 555 raises every output by 555, adding 555 to each end of the range.
−2≤y≤4 ⟶ 3≤y≤9-2 \le y \le 4 \;\longrightarrow\; 3 \le y \le 9−2≤y≤4⟶3≤y≤9
The parabola y=x2y = x^2y=x2 is shifted so its vertex is at (3,−2)(3, -2)(3,−2). Which equation is it?
Vertex (3,−2)(3, -2)(3,−2) means h=3h = 3h=3 and k=−2k = -2k=−2 in y=(x−h)2+ky = (x - h)^2 + ky=(x−h)2+k.
y=(x−3)2−2y = (x - 3)^2 - 2y=(x−3)2−2
The graph of y=(x+1)2+4y = (x + 1)^2 + 4y=(x+1)2+4 is the graph of y=x2y = x^2y=x2 shifted which way?
The inside x+1x + 1x+1 is left 111; the outside +4+4+4 is up 444.
y=(x+1)2+4 ⇒ left 1, up 4y = (x + 1)^2 + 4 \;\Rightarrow\; \text{left } 1,\ \text{up } 4y=(x+1)2+4⇒left 1, up 4
The vertex of y=x2y = x^2y=x2 is (0,0)(0, 0)(0,0). After shifting to y=(x−5)2−3y = (x - 5)^2 - 3y=(x−5)2−3, where is the vertex?
Vertex form gives vertex (h,k)(h, k)(h,k), and here h=5h = 5h=5, k=−3k = -3k=−3.
(0,0)⟶(5,−3)(0, 0) \longrightarrow (5, -3)(0,0)⟶(5,−3)
The point (3,6)(3, 6)(3,6) on y=f(x)y = f(x)y=f(x) moves to (7,6)(7, 6)(7,6) after a shift. Which shift was applied?
The height stays at 666 while the input grows from 333 to 777, so it is a horizontal shift.
7−3=4 ⇒ right 47 - 3 = 4 \;\Rightarrow\; \text{right } 47−3=4⇒right 4
The graph of g(x)=f(x)−4g(x) = f(x) - 4g(x)=f(x)−4 passes through (2,1)(2, 1)(2,1). Through which point does y=f(x)y = f(x)y=f(x) pass at x=2x = 2x=2?
Since g(x)=f(x)−4g(x) = f(x) - 4g(x)=f(x)−4, at x=2x = 2x=2 we have f(2)−4=1f(2) - 4 = 1f(2)−4=1.
f(2)=1+4=5 ⇒ (2,5)f(2) = 1 + 4 = 5 \;\Rightarrow\; (2, 5)f(2)=1+4=5⇒(2,5)
The original output is 444 higher because ggg is the graph lowered by 444.
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