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Wize High School Algebra I Textbook (Common Core) > Quadratic Functions

Vertex Form of a Quadratic Equation

Transformations of Quadratic Functions (Graphing)Previous Section
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Graphing Quadratics in Vertex Form

Recall that the graph of y=x2y=x^2 is a "U-shape" graph that passes through the origin.

Summary of Transformations

We can graph a quadratic relation in the vertex form y=a(x−h)2+ky=a(x-h)^2+k by starting with the base graph y=x2y=x^2, and applying the following transformations.

TransformationWhat happens to y=x2?Horizontal Translationh>0  →  shift right by h unitsh<0  →  shift left by h unitsVertical Stretch/Compressiona>1 or <−1  →  vertical stretch0<a<1 or −1<a<0  →  vertical compressionVertical Reflectiona<0  →  vertical reflection along the x-axisVertical Translationk>0  →  shift up by k unitsk<0  →  shift down by k units\begin{array}{|c|c|} \hline \text{Transformation}&\text{What happens to }y=x^2?\\ \hline\\ \text{Horizontal Translation}& \begin{array}{c} h>0~~\to~~\text{shift right by h units}\\ h<0~~\to~~\text{shift left by h units} \end{array}\\ \\\hline\\ \text{Vertical Stretch/Compression}& \begin{array}{c} a>1\text{ or }<-1~~\to~~\text{vertical stretch}\\ 0<a<1\text{ or }-1<a<0~~\to~~\text{vertical compression} \end{array}\\ \\\hline\\ \text{Vertical Reflection}&a<0~~\to~~\text{vertical reflection along the x-axis}\\ \\\hline\\ \text{Vertical Translation}& \begin{array}{c} k>0~~\to~~\text{shift up by k units}\\ k<0~~\to~~\text{shift down by k units} \end{array}\\ \\\hline \end{array}

Vertex

The vertex of a quadratic relation in the vertex form y=a(x−h)2+ky=a(x-h)^2+k is (h, k)(h, ~k).
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Example: Graphing Quadratics in Vertex Form

Given the quadratic relation y=−2(x+3)2+4y=-2(x+3)^2+4,
a) identify the transformations that you would apply to y=x2y=x^2 to obtain this graph.
  • Horizontal shift 3 units to the left
  • Vertical stretch by a factor of 2
  • Vertical reflection along the xx axis
  • Vertical shift 4 units up

b) identify the vertex and axis of symmetry.

The vertex is at the point (h, k)(h,~k) in the vertex formy=a(x−h)2+ky=a(x-h)^2+k.

So, the vertex is (−3, 4)(-3,~4) and the axis of symmetry is x=−3x=-3.

b) graph the quadratic relation.


Practice: Graphing Quadratics in Vertex Form

Select All of the transformations that need to be applied to the graph of y=x2y=x^2 to obtain the graph of y=−3(x−1)2−0.5y=-3(x-1)^2-0.5

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Practice: Graphing Quadratics in Vertex Form

Graph y=−12(x+1)2−54y=-\dfrac{1}{2}(x+1)^2-\dfrac{5}{4}.

Practice: Graphing Quadratics in Vertex Form

Graph the following quadratic relations and match it to its correct graph.
A.
y=(x+2)2−3y=\left(x+2\right)^2-3
B.
y=−(x−1)2+3y=-\left(x-1\right)^2+3
C.
y=−0.5(x−1)2+1.5y=-0.5\left(x-1\right)^2+1.5
D.
y=0.5(x+1)2−2y=0.5\left(x+1\right)^2-2
E.
y=(x−2)2+3y=\left(x-2\right)^2+3










Practice: Graphs of Quadratic Vertex Form

The height of a bird is given by h=−12(t−4)2+10h=-\frac{1}{2}\left(t-4\right)^2+10, where hh is the height in meters, tt is the time in seconds after the bird leaves it's nest.

a) What is the maximum height of the bird?

b) When does the bird reach its maximum height?

c) How high is the nest off the ground? (Hint: the bird leaves the nest at time t=0t=0)

d) Graph the height of the bird.
checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.

Practice: Transformations of Quadratic Graphs

The graph of y=x2y=x^2 is being transformed, where the point (−2,4)(-2,4) becomes (3,−7)(3,-7). Describe a possible set of transformations that apply in this situation and determine the new equation of the quadratic graph.