Lesson

Now that we have observed some of the key characteristics of quadratic functions, let's look at how we might create a graph of a quadratic function.

## Graphing from a table of values

Recall that you can create a graph of a function by generating a table of values and evaluating the function for certain values in its domain.  We can do the same for quadratic functions, and connect the points in a smooth curve that looks like a parabola.

#### Practice questions

##### Question 1

Consider the function $y=x^2$y=x2

1. Complete the following table of values.

 $x$x $y$y $-3$−3 $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $3$3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
2. Plot the points in the table of values.

3. Hence plot the curve.

4. Are the $y$y values ever negative?

No

A

Yes

B

No

A

Yes

B
5. Write down the equation of the axis of symmetry.

6. What is the minimum $y$y value?

7. For every $y$y value greater than $0$0, how many corresponding $x$x values are there?

$3$3

A

$1$1

B

$2$2

C

$3$3

A

$1$1

B

$2$2

C

##### Question 2

Consider the function $y=\left(x-2\right)^2$y=(x2)2

1. Complete the following table of values.

 $x$x $y$y $0$0 $1$1 $2$2 $3$3 $4$4 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
2. Sketch a graph of the function.

3. What is the minimum $y$y value?

4. What $x$x value corresponds to this minimum $y$y value?

5. What are the coordinates of the vertex? Give your answer in the form $\left(a,b\right)$(a,b).

## Graphing from characteristics in standard form

It's also possible to sketch a graph of a parabola given certain key characteristics.  We can determine these key characteristics more easily by writing the quadratic equation in standard form.

The standard form of a quadratic equation is $y=ax^2+bx+c$y=ax2+bx+c, where $a\ne0$a0.

Given a quadratic equation in standard form, we can identify certain key characteristics and use them to graph the function.

### Key characteristics

• Line of symmetry (axis of symmetry): Defined by the equation $x=\frac{-b}{2a}$x=b2a, which is a vertical line through the vertex
• Vertex (turning point): Can be found using technology or by substituting in the line of symmetry for the $x$x-value to find the $y$y-value
• $y$y-intercept: Can be found by substituting in $x=0$x=0 and evaluating
• Direction of opening: The parabola will open up if $a>0$a>0 and will open down if $a<0$a<0

Using these key characteristics, we can graph the parabola. For a better shape, we can substitute in a few $x$x-values to get a few more points.

#### Worked example

##### Question 3

Graph the quadratic function $y=x^2-6x+4$y=x26x+4

Think: We need to identify the key characteristics and then we can graph the parabola.

Do:

1. Find the line of symmetry using $x=\frac{-b}{2a}$x=b2a where $b=-6$b=6 and $a=1$a=1

 $x$x $=$= $\frac{-b}{2a}$−b2a​ $x$x $=$= $\frac{-\left(-6\right)}{2\times1}$−(−6)2×1​ $x$x $=$= $\frac{6}{2}$62​ $x$x $=$= $3$3

2. Find the y-value of the vertex by substituting in $x=3$x=3.

 $y$y $=$= $x^2-6x+4$x2−6x+4 $y$y $=$= $3^2-6\times3+4$32−6×3+4 $y$y $=$= $9-18+4$9−18+4 $y$y $=$= $-5$−5

The vertex is $\left(3,-5\right)$(3,5) with $a=1$a=1, so the parabola will open up making $\left(3,-5\right)$(3,5) a minimum.

3. Find the $y$y-intercept, but substituting in $x=0$x=0.

 $y$y $=$= $x^2-6x+4$x2−6x+4 $y$y $=$= $0^2-6\times0+4$02−6×0+4 $y$y $=$= $0-0+4$0−0+4 $y$y $=$= $4$4

So the y-intercept is $\left(0,4\right)$(0,4)

Now we put it all together on a graph.

#### Practice questions

##### Question 4

Consider the curve $y=x^2+6x+4$y=x2+6x+4.

1. Determine the axis of symmetry.

2. Hence determine the minimum value of $y$y.

3. Using the minimum point on the curve, plot the graph of the function.

##### Question 5

Consider the quadratic $y=3x^2-6x+8$y=3x26x+8

1. Find the axis of symmetry.

2. Find the vertex of the parabola. Give your answer in the form $\left(a,b\right)$(a,b).

3. What is the $y$y value of the $y$y-intercept of this quadratic function?

4. Plot the function.

### Outcomes

#### A1:F-IF.B.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

#### A1:F-IF.C.7a

Graph linear and quadratic functions and show intercepts, maxima, and minima.