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Stage 5.1-3

4.04 Transforming parabolas

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Lesson

Introduction

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Graphs of quadratic equations of the form \\y=ax^{2}+bx+c (where a,\,b,\, and c are any number and a \neq 0) are called parabolas.

The parabola defined by y=x^{2}

Parabolas will always have a y-intercept . This is the point on the graph which touches thex-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.

Transformations of parabolas

A parabola can be vertically translated by increasing or decreasing the yvalues 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. 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).

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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

Use the following applet to explore transformations of a parabola. The green dashed parabola represents y=x^{2}.

Loading interactive...

For the parabola y=a(x-b)^2+k, as a increases the parabola becomes narrower. Changing the value of k shifts the parabola vertically,and changing the value of b shifts the parabola horizontally.

Examples

Example 1

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

a

Complete the table.

x- 3- 2- 10123
y
Worked Solution
Create a strategy

Substitute each x-value into the equation.

Apply the idea

For x=-3:

\displaystyle y\displaystyle =\displaystyle 3x^{2}+2Write the equation
\displaystyle y\displaystyle =\displaystyle 3(-3)^{2}+2Substitute x=-3
\displaystyle =\displaystyle 29Evaluate

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

x- 3- 2- 10123
y29145251429
b

Plot the graph of y=3x^{2}+2.

Worked Solution
Create a strategy

Plot each point from the table and draw a curve through the points.

Apply the idea
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The points from the table are: (-3,29),\,(-2,14),\,(-1,5),\,(0,2),\,(1,5),\,(2,14), \\ (3,29).

So by plotting these points on the graph we can draw our parabola.

Example 2

Consider the parabola y=-(x+2)^{2}-5.

a

Which successive transformations turn y=x^{2} into the equation y=-(x+2)^{2}-5?

A
Reflection about the x-axis, horizontal shift 2 units to the left and vertical shift 5 units down.
B
Horizontal shift 2 units to the right, vertical shift 5 units down, reflection about the x-axis.
C
Reflection about the x-axis, vertical shift 2 units down and horizontal shift 5 units to the left.
D
Vertical shift 2 units up, horizontal shift 5 units to the left, reflection about the x-axis.
Worked Solution
Create a strategy

Consider what transformation each part of the equation represents.

Apply the idea

y=(x+2)^{2} shifts y=x^2, \, \, 2 units to the left.

y=-(x+2)^{2} reflects y=(x+2)^{2} about the x-axis.

y=-(x+2)^{2}-5 shifts y=-(x+2)^{2} down 5 units.

So the answer is option A.

b

State the coordinates of the vertex of the curve.

Worked Solution
Create a strategy

We can use the transformation from part (a).

Apply the idea
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The vertex of y=-(x+2)^{2}-5 is the vertex of y=x^{2} reflected about the x-axis, horizontally shifted 2 units to the left and shifted vertically 5 units down.

As we can see in the graph, the coordinates of the vertex of the curve are: (0-2,0-5)=\left( - 2 , - 5 \right).

c

Plot the graph of the equation.

Worked Solution
Create a strategy

Use the transformations and vertex we found in the previous parts.

Apply the idea
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Starting with the top dashed curve, y=x^2, we need to reflect the curve about the x-axis, shift the curve 2 units to the left, then shift this curve down 5 units to get the final parabola.

d

What is the axis of symmetry?

Worked Solution
Create a strategy

The axis of symmetry is the x-coodinate of the vertex.

Apply the idea

The vertex of the parabola is at (-2,-5). So the axis of symmetry is given by the equation x=-2

Example 3

Consider the equation y=(x+2)^{2}-9.

a

Find the x-intercepts.

Worked Solution
Create a strategy

Substitute y=0 into the equation and then solve for x.

Apply the idea
\displaystyle 0\displaystyle =\displaystyle (x+2)^{2}-9Substitute y
\displaystyle (x+2)^{2}\displaystyle =\displaystyle 9Move the constant term to the other side
\displaystyle x+2\displaystyle =\displaystyle \pm\sqrt{9}Take the square root of both sides
\displaystyle x+2\displaystyle =\displaystyle \pm 3 Evaluate the roots
\displaystyle x\displaystyle =\displaystyle -2 \pm 3Subtract 2 from both sides
\displaystyle =\displaystyle 1,-5Evaluate for each sign

So the x-intercepts are x=1,\,x=-5

b

Find the y-intercepts.

Worked Solution
Create a strategy

Substitute x=0 into the equation and then solve for y.

Apply the idea
\displaystyle y\displaystyle =\displaystyle (0+2)^{2}-9Substitute x=0
\displaystyle =\displaystyle -5Evaluate
c

Determine the coordinates of the vertex.

Worked Solution
Create a strategy

Consider the transformations that have occurred.

Apply the idea

y=(x+2)^{2} is y=x^{2} shifted 2 units to the left and 9 units down.

The coordinates of the vertex are (0-2,0-9)=(-2,\,-9).

d

Plot the graph of the equation.

Worked Solution
Create a strategy

Plot the vertex and intercepts, and sketch the parabola through these points.

Apply the idea
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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

MA5.2-10NA

connects algebraic and graphical representations of simple non-linear relationships

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