10. Quadratics

Lesson

A quadratic expression is a type of polynomial that has a degree of $2$2. A quadratic equation can appear in many different forms as we will see in later lessons. If manipulated, all quadratic equations can be written in the form $y=ax^2+bx+c$`y`=`a``x`2+`b``x`+`c`, where $a\ne0$`a`≠0.

See the examples and non-examples of quadratic expressions below:

Examples |
---|

$5x^2$5x2 |

$\frac{x^2}{6}+3x-7$x26+3x−7 |

$2x+7-18x^2$2x+7−18x2 |

$-2\left(x-3\right)^2+7$−2(x−3)2+7 |

Nonexamples | Reason |
---|---|

$\frac{1}{x}+4$1x+4 |
Not a polynomial as $\frac{1}{x}=x^{-1}$1x=x−1 |

$\sqrt{x+2}-7$√x+2−7 |
Not a polynomial as there is a square root |

$x^3+x^2$x3+x2 |
Has a degree of $3$3, not $2$2 |

The shape of a quadratic function is called a parabola. The graph of a quadratic equation is a curve with one line of symmetry and one vertex (turning point). All quadratic equations have common properties, apart from just having a degree of $2$2. Use this interactive to explore different quadratic equations. Can you see the common physical properties?

At this stage, all we really want to explore is the shape of the function.

Is $7x^3+3x^2$7`x`3+3`x`2 a quadratic expression?

Yes

ANo

BYes

ANo

B

Which of the following equations represent a quadratic relationship between $x$`x` and $y$`y`?

$y=1-x^2$

`y`=1−`x`2A$y=\left(x^2-1\right)^3$

`y`=(`x`2−1)3B$y=x^2+\frac{2}{x}$

`y`=`x`2+2`x`C$y=\sqrt{x^2+2}$

`y`=√`x`2+2D$y=2x^2$

`y`=2`x`2E$y=1-x^2$

`y`=1−`x`2A$y=\left(x^2-1\right)^3$

`y`=(`x`2−1)3B$y=x^2+\frac{2}{x}$

`y`=`x`2+2`x`C$y=\sqrt{x^2+2}$

`y`=√`x`2+2D$y=2x^2$

`y`=2`x`2E

What relationship between $x$`x` and $y$`y` is represented by the given graph?

Loading Graph...

Linear

AQuadratic

BOther

CLinear

AQuadratic

BOther

C

In previous chapters, we learned how to find certain features of linear equations including the slope, the $x$`x`-intercept, and the $y$`y`-intercept. Now we are going to explore the features of parabolas.

The intercepts are where the parabola crosses the horizontal ($x$`x`) and vertical ($y$`y`) axes.

More specifically:

- The $x$
`x`-intercepts are where the parabola crosses the $x$`x`-axis. This occurs when $y=0$`y`=0. - The $y$
`y`-intercept is where the parabola crosses the $y$`y`-axis. This occurs when $x=0$`x`=0.

You can see them in the picture below. Note, however, that there won't always be two $x$`x`-intercepts. Sometimes there may only be one or even none.

For now, we will just identify the intercepts from a graph, later we will look at doing it algebraically.

Notice that the $y$`y`-intercept is where the function crosses the $y$`y`-axis in the image above. We can write this as $y=-3$`y`=−3 or $\left(0,-3\right)$(0,−3).

We can also see from this image that the $x$`x`-intercepts are at $x=-1$`x`=−1 and $x=3$`x`=3. We can also write these as $\left(-1,0\right)$(−1,0) and $\left(3,0\right)$(3,0).

Maximum or minimum values are found at the turning points, or the vertex of the parabola.

Parabolas that have a positive leading term open up and have a minimum value. This means the $y$`y`-value will never go under a certain value.

Parabolas that have a negative leading term open down and have a maximum value. This means the $y$`y`-value will never go over a certain value.

Maximum and minimum values occur on a parabola's line of symmetry. This is the line that evenly divides a parabola into two sides down the middle.

- Determine whether the parabola opens up or down so we know whether we're finding a minimum or maximum.
- From the graph, the turning point (vertex) will be the point on the line of symmetry where the parabola changes from increasing to decreasing or vice versa - it should be stated as coordinates $x,y$
`x`,`y`. - The minimum or maximum value is referring specifically to the $y$
`y`-value of the turning point. This will also help us to state the range of the function. - The intervals of increase or decrease are referring to the domain of $x$
`x`-values over which the parabola is increasing or decreasing, the boundary will be the $x$`x`-value of the turning point.

Examine the given graph and answer the following questions.

Loading Graph...

What are the $x$

`x`values of the $x$`x`-intercepts of the graph? Write both answers on the same line separated by a comma.What is the $y$

`y`value of the $y$`y`-intercept of the graph?What is the minimum value of the graph?

Examine the attached graph and answer the following questions.

Loading Graph...

What is the $x$

`x`-value of the $x$`x`-intercept of the graph?What is the $y$

`y`value of the $y$`y`-intercept of the graph?What is the absolute maximum of the graph?

Determine the interval of $x$

`x`in which the graph is increasing.Give your answer as an inequality.

Consider the graph of the function $y=f\left(x\right)$`y`=`f`(`x`) and answer the following questions.

Loading Graph...

What is the $y$

`y`-value of the absolute minimum of the graph?Hence determine the range of the function.

$y\ge\editable{}$

`y`≥What is the domain of this function?

All real $x$

`x`A$x\ge-4$

`x`≥−4B$x\le-2$

`x`≤−2C$x>-2$

`x`>−2DAll real $x$

`x`A$x\ge-4$

`x`≥−4B$x\le-2$

`x`≤−2C$x>-2$

`x`>−2D

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