12 multiple-choice questions, progressively harder.
How is the graph of y=3f(x)y = 3f(x)y=3f(x) related to the graph of y=f(x)y = f(x)y=f(x)?
Solution
Correct answer: C
The 333 multiplies the output from outside the function, so it scales every height. A factor greater than 111 stretches the graph vertically.
y=3f(x) ⇒ vertical stretch by 3y = 3f(x) \;\Rightarrow\; \text{vertical stretch by } 3y=3f(x)⇒vertical stretch by 3
How is the graph of y=−f(x)y = -f(x)y=−f(x) related to the graph of y=f(x)y = f(x)y=f(x)?
Correct answer: A
The minus sign negates every output, sending each height to its opposite.
y=−f(x) ⇒ reflection across the x-axisy = -f(x) \;\Rightarrow\; \text{reflection across the x-axis}y=−f(x)⇒reflection across the x-axis
Negating the output flips the graph over the x-axis.
How is the graph of y=12f(x)y = \tfrac{1}{2}f(x)y=21f(x) related to the graph of y=f(x)y = f(x)y=f(x)?
Correct answer: B
The factor 12\tfrac{1}{2}21 multiplies each output, and a factor between 000 and 111 shrinks every height.
y=12f(x) ⇒ vertical compression by 2y = \tfrac{1}{2}f(x) \;\Rightarrow\; \text{vertical compression by } 2y=21f(x)⇒vertical compression by 2
How is the graph of y=f(−x)y = f(-x)y=f(−x) related to the graph of y=f(x)y = f(x)y=f(x)?
Correct answer: D
The minus sign is on the input, so it flips the sign of every xxx-position while keeping heights.
y=f(−x) ⇒ reflection across the y-axisy = f(-x) \;\Rightarrow\; \text{reflection across the y-axis}y=f(−x)⇒reflection across the y-axis
How is the graph of y=f(2x)y = f(2x)y=f(2x) related to the graph of y=f(x)y = f(x)y=f(x)?
The 222 multiplies the input, so it scales every xxx-position by 12\tfrac{1}{2}21, pulling points toward the y-axis.
2x=a ⇒ x=a22x = a \;\Rightarrow\; x = \tfrac{a}{2}2x=a⇒x=2a
A factor greater than 111 inside compresses the graph horizontally.
The point (4,3)(4, 3)(4,3) is on the graph of y=f(x)y = f(x)y=f(x). Which point must be on the graph of y=2f(x)y = 2f(x)y=2f(x)?
Multiplying by 222 outside is a vertical stretch, so keep the input and double the height.
(4, 2×3)=(4,6)(4,\; 2 \times 3) = (4, 6)(4,2×3)=(4,6)
The point (6,2)(6, 2)(6,2) is on the graph of y=f(x)y = f(x)y=f(x). Which point must be on the graph of y=12f(x)y = \tfrac{1}{2}f(x)y=21f(x)?
The factor 12\tfrac{1}{2}21 halves the height and leaves the input unchanged.
(6, 12×2)=(6,1)(6,\; \tfrac{1}{2} \times 2) = (6, 1)(6,21×2)=(6,1)
Which parabola opens downward?
A negative coefficient reflects the bowl across the x-axis, turning it upside down.
y=−x2 ⇒ opens downwardy = -x^2 \;\Rightarrow\; \text{opens downward}y=−x2⇒opens downward
Every positive coefficient keeps the bowl opening upward.
Which rule reflects the graph of y=f(x)y = f(x)y=f(x) across the x-axis?
Reflecting across the x-axis negates the output, which is a minus sign outside the function.
y=−f(x)y = -f(x)y=−f(x)
A minus inside, f(−x)f(-x)f(−x), would reflect across the y-axis instead.
The point (−3,4)(-3, 4)(−3,4) is on the graph of y=f(x)y = f(x)y=f(x). Which point must be on the graph of y=f(−x)y = f(-x)y=f(−x)?
Reflecting across the y-axis flips the sign of the input and keeps the height.
(−3, 4)⟶(3,4)(-3,\; 4) \longrightarrow (3, 4)(−3,4)⟶(3,4)
Which parabola is wider (flatter) than y=x2y = x^2y=x2?
A stretch factor between 000 and 111 makes a parabola climb slowly, so it looks wider. Only 14\tfrac{1}{4}41 is less than 111 in size.
y=14x2 ⇒ wider than y=x2y = \tfrac{1}{4}x^2 \;\Rightarrow\; \text{wider than } y = x^2y=41x2⇒wider than y=x2
The factor −1-1−1 has size 111, so y=−x2y = -x^2y=−x2 is the same width, just flipped.
The point (10,6)(10, 6)(10,6) is on the graph of y=f(x)y = f(x)y=f(x). Which point must be on the graph of y=f(2x)y = f(2x)y=f(2x)?
A point reappears where the inside 2x2x2x equals the old input 101010.
2x=10 ⇒ x=52x = 10 \;\Rightarrow\; x = 52x=10⇒x=5
So (10,6)(10, 6)(10,6) moves to (5,6)(5, 6)(5,6), with the height unchanged.
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