Preparing for this section

  • Intercepts
  • Completing the Square
  • Quadratic Equations
  • Graphing Techniques


1. Finding Vertex and Axis of Symmetry
2. Graphing Quadratic Functions
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.

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 $Vertex=( -\dfrac{b}{2a},ƒ(- \dfrac{b}{2a}))$.
$x=- \dfrac{b}{2a}=- \dfrac{-3}{2(4)}=\dfrac{3}{8}=0.375$
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.

2. Graphing Quadratic Functions

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.
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

The y-intercept is always where the graph crosses the y-axis, which means x=0.

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


The x-intercept is always where the graph crosses the x-axis, which means y=0

This does not factor, so let's try using the quadratic formula:



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

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?
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.

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.
$Max.Income = (100-2x)(0.8+0.05x)$
$Max.Income = (100-34)(0.8+0.85)$
$Max.Income = (66)(1.65)$
$Max.Income = 108.9$

The answer is $108.9

Skill building

  1. For each of the following quadratic functions find the vertex, axis of symmetry, whether the parabola opens up or down and sketch it:
    1. $y = x^2-2x+2$
    2. $y = 2x^2-x+6$
    3. $y = 4x^2-2x+8$
    4. $y = -4x^2-4x+1$
    5. $y = 0.2x^2-0.1x+0.0125$
    6. $y = x(x+5)-10$
  2. Using quadratic formula, find solutions to each of the following equations
    1. $x^2-4x+2 = 0$
    2. $x^2 = 10x-15$
    3. $x(x+5) = -10$
    4. $-x^2+31x-50 = 0$


  1. The answers are in order: vertex, axis of symmetry, up(down).
    1. (1;1); 1; up.
    2. (0.25;5.875); 0.25; up.
    3. (0.25;7.75); 0.25; up.
    4. (-0.5;2); -0.5; down.
    5. (0.25;0); 0.25; up.
    6. (2.5;8.75); 2.5; up.
  2. Rounded to hundredth
    1. 0.58 and 3.41
    2. 1.84 and 8.16
    3. no solution
    4. 29.29 and 1.7