### Preparing for this section

• Intercepts
• Completing the Square
• Graphing Techniques

### Objectives

1. Finding Vertex and Axis of Symmetry
3. Using Desmos to Find the Quadratic Function of Best Fit to Data

A quadratic function is one of the form ƒ(x) = ax2+ bx + c, where a, b, and c are numbers with a not equal to zero.

Quadratic functions are widely used in science, business and engineering. The U-shape of a parabola can describe the trajectories of a bouncing ball and of water jets in a fountain. Most of the objects we use every day, from cars to clocks, would not exist if someone, somewhere hadn't applied quadratic functions to their design.

Here’s an example: you dive from a platform 3 meters high into the water. Let’s say you hit the water 4 meters away from the platform. That’s how your path is going to look like:

##### 1. Finding Vertex and Axis of Symmetry

$ƒ(x)=ax^2+bx+c, a≠0$
$Vertex=( -\dfrac{b}{2a},ƒ(- \dfrac{b}{2a}))$
Parabola opens up if a>0; vertex is a minimum point.
Parabola opens down if a<0; vertex is a maximum point.
Axis of symmetry: $x = -\dfrac{b}{2a}$

Example 1
Find the coordinates of the vertex of $ƒ\left(x\right)=4x^2-3x+2$. Without graphing, determine if the vertex is the maximum or minimum point of the quadratic function.

Solution
This quadratic function is written in the form $ƒ(x)=ax^2+bx+c, a≠0$. That means we can find the vertex using the formula $Vertex=( -\dfrac{b}{2a},ƒ(- \dfrac{b}{2a}))$.
$a=4,b=-3,c=2$
$x=- \dfrac{b}{2a}=- \dfrac{-3}{2(4)}=\dfrac{3}{8}=0.375$
$ƒ(-\dfrac{b}{2a})=ƒ(\dfrac{3}{8})=0.5625-1.125+2=1.4375$
The Vertex would be $(0.375;1.4375)$.

Axis of Symmetry:
$x = -\dfrac{b}{2a}$
$x = -\dfrac{-3}{2(4)}$
$x = 0.375$
Next we need to determine if the vertex that we found is a maximum or a minimum point, without graphing. Knowing which direction the curve opens can help us answer this question. Since a=4, and 4 is more than 0, this parabola would open up. That means that the vertex is a minimum point.

From the information we got in Example 1 we can graph $ƒ(x)=ax^2+bx+c, a≠0$, by hand.
To graph a quadratic function we need to follow those 4 steps

1. Does the graph open up or down?
2. Find the Vertex
3. Find the Intercepts
4. Graph the Parabola

Example 2
Use the vertex and the intercepts to graph the function. Find the equation for this function's axis of symmetry.
Solution
Step 1. Does the Graph Open Up or Down?
a = 4 and 4 > 0, so as we already found out that this parabola opens up.

Step 2. Find the Vertex
To find the vertex we can use the formula: $Vertex=( -\dfrac{b}{2a},ƒ(- \dfrac{b}{2a}))$
But since we know that our vertex is $(0.375;1.4375)$, we can move on to the next step.
Step 3. Find the Intercepts
y-intercept

The y-intercept is always where the graph crosses the y-axis, which means x=0.
$ƒ\left(x\right)=4x^2-3x+2$
$ƒ\left(0\right)=4(0)^2-3(0)+2$

The y-intercept is $(0;2)$

x-intercept

The x-intercept is always where the graph crosses the x-axis, which means y=0
$ƒ\left(x\right)=4x^2-3x+2$
$0=4x^2-3x+2$

This does not factor, so let's try using the quadratic formula:
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$x=\frac{-4\pm\sqrt{(-3)^2-4(4)(2)}}{2(4)}$

$x=\frac{-4\pm\sqrt{-23}}{8}$

The number under the square root is negative so there is no real number solution. Also there are NO x-intercepts.

Step 4. Graph the parabola

$ƒ\left(x\right)=4x^2-3x+2$
y-int=(0;2)
x-int=none
vertex:(0.375;1.4375)
axis of symmetry: x=0.375
a = 4 > 0, opens up

##### 3. Using Desmos to Find the Quadratic Function of Best Fit to Data

Example 3
The marketer sells around 100 bananas a day. The cost of one banana is $0.80. The marketer counted that for every$0.05 price increase 2 fewer bananas will be sold. Graph and determine what cost will maximize the profit?
Solution
Let's say x is the number of 0.05 price increases.
Then 0.8+0.05x is the single price,
so 100-2x is the number of bananas sold.

Now, to find the income we need to multiply the number of bananas sold by the price.
$Income=(100-2x)(0.8+0.05x)$
$Income=-0.1x^2+3.4x+80$

We got a quadratic function. a=-0.1, which means that the parabola opens down, so x is the number of prices increases needed to maximize the profit.
To find x we will use the formula:
$x = -\dfrac{b}{2a}$
$x = -\dfrac{3.4}{2(-0.1)}$
$x = 17$

This means that the profit will be maximized after 17 $0.05 price increases. So one banana will cost:$Banana = 0.8+0.05(17)Banana = 1.65$The most profitable price of one banana is$1.65.

Now we can find the maximum income.
$Max.Income = (100-2x)(0.8+0.05x)$
$Max.Income = (100-34)(0.8+0.85)$
$Max.Income = (66)(1.65)$
$Max.Income = 108.9$