10. Quadratics

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.

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.

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

Complete the following table of values.

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

Loading Graph...Hence plot the curve.

Loading Graph...Are the $y$

`y`values ever negative?No

AYes

BWrite down the equation of the axis of symmetry.

What is the minimum $y$

`y`value?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

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

Complete the following table of values.

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

Loading Graph...What is the minimum $y$

`y`value?What $x$

`x`value corresponds to this minimum $y$`y`value?What are the coordinates of the vertex? Give your answer in the form $\left(a,b\right)$(

`a`,`b`).

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`=`a``x`2+`b``x`+`c`, where $a\ne0$`a`≠0.

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

- Line of symmetry (axis of symmetry): Defined by the equation $x=\frac{-b}{2a}$
`x`=−`b`2`a`, 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.

Graph the quadratic function $y=x^2-6x+4$`y`=`x`2−6`x`+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`=−`b`2`a` 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.

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

Determine the axis of symmetry.

Hence determine the minimum value of $y$

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

Loading Graph...

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

Find the axis of symmetry.

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

`a`,`b`).What is the $y$

`y`value of the $y$`y`-intercept of this quadratic function?Plot the function.

Loading Graph...

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

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