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Wize High School Grade 10 Math Textbook > Vertex Form of a Quadratic Equation

Translations (Shifts)

Vertical Stretches and ReflectionsPrevious Section
Graphing Quadratics in Vertex FormNext Section
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Translations (Shifts)

Recall that the vertex form of a quadratic equation is y=a(x−h)2+ky=a\left(x-h\right)^2+k. Let's explore how these different numbers a, h, and ka,\ h,\ \text{and}\ k affect the graph.

The graph of y=x2+k\bm{\colorOne{y=x^2+k}}

Based on the following graphs, try to identify the role kk plays in graphing a quadratic graph.

Write it Down
How does the kk value affect the graph of y=x2+k\bm{y=x^2+k}?
  • If kk is positive, the graph is
    shifted (translated) up k units
  • If kk is negative, the graph is
    shifted (translated) down k units
The transformation is
a vertical translation up or down

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The graph of y=(x−h)2\bm{\colorOne{y=(x-h)^2}}

Based on the following graphs, try to identify the role hh plays in graphing a quadratic graph.


Write it Down
How does the hh value affect the graph of y=(x−h)2\bm{y=(x-h)^2}?
  • If hh is positive, the graph is
    shifted (translated) h units to the right
  • We will see a negative number within the brackets like y=(x−3)2y=(x-3)^2
  • If hh is negative, the graph is
    shifted (translated) h units to the left
  • We will see a positive number within the brackets like y=(x+3)2y=(x+3)^2
The transformation is
a horizontal translation right or left

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Example: Translations

Determine the equation of the orange graph above.

The graph is shifted 2 units to the right and 3 units up, so the equation is y=(x+2)2+3y=(x+2)^2+3.

Practice: Translations

Determine the equation of the graph that results in applying the following transformations to the graph of y=x2y=x^2.

a) Vertical shift up by 3 units

b) Vertical shift down by 2 units

c) Horizontal shift to the right by 5 units

d) Horizontal shift to the left by 4 units

Practice: Translations

Determine the equation of the following graphs.

a)

b)

c)