Question

What kind of transformation converts the graph of f(x)=4x^2–10 into the graph of g(x)=x^2–10?
vertical stretch
horizontal stretch
horizontal shrink
vertical shrink
reflection across the y-axis
reflection across the x-axis

Answer 1

Its horizontal stretch.
Remember:
Transformation of f(x) Transformation of graph of f(x)
af(x), a>1 Stretch vertically
af(x), 0<a<1 Shrink vertically
f(ax), a>1 Shrink horizontally
f(ax), 0<a<1 Stretch horizontally
The vertex form of a quadratic function is f(x)=m(x–h)^2+k. The vertex is (h,k).
Rewrite the function rule for g(x) as a transformation of the rule for f(x). That is, write g(x)=x^2–10 as af(x) or f(ax) since all the answer choices describe transformations of one of those forms.
To find the specific transformation, write g(x), af(x), and f(ax) in vertex form. Then, solve for the values of a so that g(x)=af(x), and g(x)=f(ax). If there is no such value of a for one of the equations, then g(x) is not a transformation of that type of f(x).
Write the functions in vertex form.
Write g(x) in vertex form.
g(x) = x^2–10
= 1(x–0)^2+–10
Write af(x) in vertex form.
af(x) = a(4x^2–10)
= 4a(x–0)^2+–10a
Write f(ax) in vertex form.
f(ax) = 4(ax)^2–10
= 4[a(x–0)]^2–10
= 4a^2(x–0)^2+–10
Set g(x)=af(x) and solve for a.
g(x) = af(x)
1(x–0)^2+–10 = 4a(x–0)^2+–10a
The functions g(x) and af(x) are equal if 1=4a, 0=0, and
–10 = –10a. Solve these equations for a.
Solve 1=4a:
1 = 4a
1/4 = a
Solve –10 = –10a:
–10 = –10a
1 = a
Since 1/4 ≠ 1, g(x) cannot equal af(x), so the transformation is not of the form f(x)↦af(x).
Set g(x)=f(ax) and solve for a.
g(x) = f(ax)
1(x–0)^2+–10 = 4a^2(x–0)^2+–10
The functions g(x) and f(ax) are equal if 1=4a^2, 0=0, and –10 = –10. Solve these equations for a.
Solve 1=4a^2:
1 = 4a^2
1/4 = a^2
±1/2 = a
Since a = ±1/2 is a solution to this equation, g(x)=f(ax) for a = ±1/2. This means that both of the transformations f(x)↦f(–1/2x) and f(x)↦f(1/2x) convert f to g. The first does not correspond to any of the answer choices, but the second is a horizontal stretch since 0<1/2<1.

Answer 2

Vertical shrink — g(x) = (1/4)f(x), so the graph is vertically shrunk by a factor of 1/4 (heights reduced to one quarter).