4 Quadratics and polynomials

Lesson

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.

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.

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}.

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.

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).

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

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.

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

a

Complete the table.

x | - 3 | - 2 | - 1 | 0 | 1 | 2 | 3 |
---|---|---|---|---|---|---|---|

y |

Worked Solution

b

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

Worked Solution

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

b

State the coordinates of the vertex of the curve.

Worked Solution

c

Plot the graph of the equation.

Worked Solution

d

What is the axis of symmetry?

Worked Solution

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

a

Find the x-intercepts.

Worked Solution

b

Find the y-intercepts.

Worked Solution

c

Determine the coordinates of the vertex.

Worked Solution

d

Plot the graph of the equation.

Worked Solution

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}

Explore the connection between algebraic and graphical representations of relations such as simple quadratic, reciprocal, circle and exponential, using digital technology as appropriate