Like lines, parabolas will always have a y-intercept . This is the point on the graph which touches the y-axis. We can find this by setting x=0 and finding the value of y.

Similarly, we can look for x-intercepts by setting y=0 and then solving for x. Because this is a quadratic equation, there could be 0,\, 1, or 2 solutions, and there will be the same number of x-intercepts.

Parabolas have an axis of symmetry which is the vertical line x=-\dfrac{b}{2a}. This is also the midpoint of the x-intercepts if they exist.

The point on the parabola which intersects the axis of symmetry is called the vertex of the parabola. The x-value of the vertex will be the axis of symmetry, and we can find the y-value by substituting this x-value into the equation.

Finally, parabolas have a concavity. If the vertex is the minimum point on the graph then the parabola is concave up and if the vertex is the maximum point on the graph then the parabola is concave down.

Examples

Example 1

Consider the equation y = - x^{2}.

Complete the following table of values.

x	- 3	- 2	- 1	0	1	2	3
y

Worked Solution

Create a strategy

Substitute the values from the table into the equation.

Apply the idea

For x=-3:

\displaystyle y	\displaystyle =	\displaystyle -x^2
	\displaystyle =	\displaystyle -(-3)^2	Substitute -3 to x
	\displaystyle =	\displaystyle -(9)	Square -3
	\displaystyle =	\displaystyle -9	Evaluate

Similarly, by substituting the remaining x-values into y=-x^2, we get:

x	- 3	- 2	- 1	0	1	2	3
y	-9	-4	-1	0	-1	-4	-9

Plot the points in the table of values.

Worked Solution

Create a strategy

Plot the points from the completed table in part (a).

Apply the idea

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The ordered pairs of points to be plotted on the coordinate plane are (-3,-9), (-2,-4), (-1,-1), (0,0), (1,-1), (2,-4) and (3,-9), which are plotted on the graph.

Sketch the curve.

Worked Solution

Create a strategy

Draw a curve through the plotted points on the coordinate plane from part (b).

Apply the idea

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Are the y-values ever positive?

Worked Solution

Create a strategy

Use the table of values from part (a) and the graph from part (c).

Apply the idea

From the completed table of values in part (a), we can see that y-values are all negative or 0. Since the graph formed in part (c) is also concave down any other y-values will be further in the negative direction.

So y-values are never positive.

What is the maximum y-value?

Worked Solution

Create a strategy

Get the y-coordinate of the highest point on the graph.

Apply the idea

In part (c), the highest point on the graph is (0,0).

So the maximum y-value is y=0.

Write down the equation of the axis of symmetry.

Worked Solution

Create a strategy

The x-value of the vertex will be the axis of symmetry.

Apply the idea

The vertex is the minimum or maximum point on the graph. In part (e), the maximum point on the graph is (0,0). So (0,0) is the vertex of the parabola.

So the equation of the axis of symmetry isx=0

Idea summary

The graph of a quadratic equation of the form y=ax^{2}+bx+c is a parabola.

Parabolas have a y-intercept and can have 0,\, 1,or 2 x-intercepts, depending on the solutions to the quadratic equation.

Parabolas have a vertical axis of symmetry and a vertex which is the point on the graph which intersects the axis of symmetry.

Parabolas are either concave up or concave down, depending on whether the vertex is the minimum or maximum point on the graph.

Transformations of parabolas

A parabola can be vertically translated by increasing or decreasing the y-values by a constant number. So to translate y=x^{2} up by k units gives us y=x^{2} + k.

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This graph shows y=x^2 translated vertically up by 2 to get y=x^{2} + 2, and down by 2 to get y=x^{2} - 2.

Similarly, a parabola can be horizontally translated by increasing or decreasing the x-values by a constant number. However, the x-value together with the translation must be squared together. That is, to translate y=x^{2} to the left by h units we get y=(x+h)^{2}.

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This graph shows y=x^2 translated horizontally left by 2 to get y=(x+2)^{2} and right by 2 to gety=(x-2)^{2}.

A parabola can be vertically scaled by multiplying every y-value by a constant number. So to expand the parabola y=x^{2} by a scale factor of a we get y=ax^{2}. We can compress a parabola by dividing by the scale factor instead.

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This graph shows y=x^2 vertically expanded by a scale factor of 2 to get y=2x^{2} and compressed by a scale factor of 2 to get y=\dfrac{x^{2}}{2}.

Finally, we can reflect a parabola about the x-axis by taking the negative.

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So to reflect y=x^{2} about the x-axis gives us y=-x^{2}. Notice that reflecting will change the concavity (in this case from concave up to concave down).

Exploration

The following applet demonstrates how a scale factor affects the shape of a parabola. Play with the applet below by dragging the sliders.

Loading interactive...

When a>1, the function is expanded vertically. If a<1, it is compressed vertically, and if a is negative, it is reflected across the x-axis.

Examples

Example 2

Consider the equation y = 3x^{2}.

Complete the following table of values.

x	- 3	- 2	- 1	0	1	2	3
y						12	27

Worked Solution

Create a strategy

Substitute the values from the table into the equation.

Apply the idea

For x=-3:

\displaystyle y	\displaystyle =	\displaystyle 3x^2	Write the equation
	\displaystyle =	\displaystyle 3(-3)^2	Substitute x=-3
	\displaystyle =	\displaystyle 27	Evaluate

Similarly, by substituting the remaining x-values into y=3x^2, we get:

x	- 3	- 2	- 1	0	1	2	3
y	27	12	3	0	3	12	27

Graph y=3x^2.

Worked Solution

Create a strategy

Plot the points from the table of values and draw the curve through each plotted point.

Apply the idea

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The ordered pairs of points are (-3,27),(-2,12),(-1,3),(0,0),(1,3),(2,12) and (3,27).

This curve of y=3x^2 must pass through each of the plotted points.

Example 3

Consider the equation y = \left(x - 2\right)^{2}.

Complete the following table of values.

x	0	1	2	3	4
y

Worked Solution

Create a strategy

Substitute the values from the table into the equation.

Apply the idea

For x=0:

\displaystyle y	\displaystyle =	\displaystyle \left(x - 3\right)^{2}	Write the equation
	\displaystyle =	\displaystyle \left(0 - 2\right)^{2}	Substitute 0 to x
	\displaystyle =	\displaystyle \left(-2\right)^{2}	Evaluate the subtraction
	\displaystyle =	\displaystyle 4	Evaluate

Similarly, by substituting the remaining x-values into \left(x - 2\right)^{2}, we get:

x	0	1	2	3	4
y	4	1	0	1	4

Sketch the parabola.

Worked Solution

Create a strategy

Plot the points in the table of values and draw the curve passing through each plotted point.

Apply the idea

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The ordered pairs of points to be plotted on the coordinate plane are (0,4),(1,1),(2,0),(-1,1) and (-2,4).

The parabola \left(x - 2\right)^{2} must pass through each of the plotted points.

What is the minimum y-value?

Worked Solution

Create a strategy

Get the y-coordinate of the lowest point on the graph.

Apply the idea

In part (b), the lowest point on the graph is (2,0).

So the minimum y-value is y=0.

What x-value corresponds to this minimum y-value?

Worked Solution

Create a strategy

Get the x-coordinate of the lowest point on the graph.

Apply the idea

In part (b), the lowest point on the graph is (2,0).

So the x-value that corresponds to the minimum y-value is x=2.

What are the coordinates of the vertex?

Worked Solution

Create a strategy

The vertex here is the lowest point on the curve.

Apply the idea

In part (b), the lowest point on the graph is (2,0).

So the coordinates of the vertex are (2,0).

Idea summary

Parabolas can be transformed in the following ways (starting with the parabola defined by y=x^{2}):

Vertically translated by k units: y=x^{2} + k
Horizontally translated by h units: y=(x-h)^{2}
Vertically scaled by a scale factor of a: y=ax^{2}
Vertically reflected about the x-axis: y=-x^{2}

Outcomes

VCMNA311

Graph simple non-linear relations with and without the use of digital technologies and solve simple related equations

5.08 Graphs of parabolas

Ideas

Features of parabolas

Examples

Example 1

Transformations of parabolas

Exploration

Examples

Example 2

Example 3

Outcomes

VCMNA311

What is Mathspace

About Mathspace