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10.05 Graphing quadratic functions

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

    Loading Graph...

  3. Hence plot the curve.

    Loading Graph...

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

    Loading Graph...

  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 $=$= $x^2-6x+4$x26x+4
$y$y $=$= $3^2-6\times3+4$326×3+4
$y$y $=$= $9-18+4$918+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 $=$= $x^2-6x+4$x26x+4
$y$y $=$= $0^2-6\times0+4$026×0+4
$y$y $=$= $0-0+4$00+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.

    Loading Graph...

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.

    Loading Graph...

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.

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