Topics
7. Quadratic Functions
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United States of AmericaFL
Grade 912

7.04 Quadratic functions in vertex form

Lesson
Practice
Lesson

Concept summary

One way to represent quadratic functions is using vertex form. This form allows us to identify the coordinates of the vertex of the parabola, as well as the direction of opening and scale factor that compresses or stretches the graph of the function.

\displaystyle f\left(x\right) = a\left(x - h\right)^2 + k
\bm{\left(h, k\right)}
The coordinates of the vertex.
\bm{a}
The scale factor, which tells us about the shape of the graph.

If we know the coordinates of the vertex, we actually only need to know one other point on the graph, such as the y-intercept, to be able to draw the graph of quadratic function. As the parabola is symmetric across the line of symmetry, which the vertex lies on, we can use the properties of symmetry to find a third point on the graph.

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As an example, consider the graph of \\y=\left(x-2\right)^2-3

  • Vertex at \left(2, -3\right)
  • Axis of symmetry x=2
  • y-intercept at \left(0,1\right) has a corresponding point at \left(4, 1\right)

Worked examples

Example 1

The table of values below represents a quadratic function. Write the function p(x) in vertex form.

x-4-3-2-1012
p(x)-503430-4

Approach

We can use the fact that a quadratic function has symmetry about its vertex to identify the location of the vertex from the table.

Solution

Looking at the values of p\left(x\right), we can see that it has a maximum value of 4 and falls off symmetrically on either side. So we know that the vertex is the point \left(-1, 4\right), and therefore the quadratic is of the form p\left(x\right) = a\left(x + 1\right)^2 + 4

We can now find the value of a by substituting any other pair of values from the table, such as \left(0, 3\right). Doing so, we get 3 = a\left(0 + 1\right)^2 + 4which we can solve to get a = -1.

So the quadratic function shown in the table of values is p\left(x\right) = -\left(x + 1\right)^2 + 4

Example 2

The quadratic function f\left(x\right) = 2x^2 has been transformed to produce a new quadratic function g\left(x\right), as shown in the graph:

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a

Describe the transformation from f\left(x\right) to g\left(x\right).

Solution

The function g\left(x\right) has the same shape and size as f\left(x\right), but has been shifted to the left. Comparing the vertices of the two parabolas, we can see that this is a translation of 6 units to the left.

b

Write the equation of the function g\left(x\right) in vertex form.

Approach

Remember that vertex form for a quadratic is g\left(x\right) = a\left(x - h\right)^2 + k, where the vertex is at the point \left(h, k\right).

Solution

f\left(x\right) = 2x^2 has been translated 6 units to the left to produce g\left(x\right), and we can see that its vertex is at \left(-6, 0\right). So g\left(x\right) has can be written as g\left(x\right) = 2\left(x + 6\right)^2.

Example 3

Consider the quadratic function A\left(x\right) = 5 - \left(x - 3\right)^2.

a

Determine the coordinates of the vertex.

Approach

A quadratic function in the form A\left(x\right) = a\left(x - h\right)^2 + k has a vertex at \left(h, k\right).

Solution

In this case the quadratic is A\left(x\right) = -\left(x - 3\right)^2 + 5, and so the vertex is at \left(3, 5\right).

b

Determine the equation of the axis of symmetry.

Approach

The axis of symmetry of a parabola is the vertical line x = h, which passes through its vertex.

Solution

We found that the vertex is the point \left(3, 5\right), and therefore the axis of symmetry is the line x = 3.

c

Determine the y-intercept.

Approach

The y-intercept of a function is the point on the function at which x = 0.

Solution

Substituting x = 0 into the function, we have y = A\left(0\right) = 5 - \left(0 - 3\right)^2

Evaluating this, we find that y = -4 and so the y-intercept is the point \left(0, -4\right).

d

Draw the graph of the parabola.

Approach

We have found the vertex, axis of symmetry, and y-intercept of this parabola. We can use these features to help draw the graph.

Solution

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Outcomes

MA.912.AR.3.7

Given a table, equation or written description of a quadratic function, graph that function, and determine and interpret its key features.

MA.912.F.2.1

Identify the effect on the graph or table of a given function after replacing f(x) by f(x)+k,kf(x), f(kx) and f(x+k) for specific values of k.

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