About Graphing Quadratic Functions

Quadratic function has the form $ f(x) = ax^2 + bx + c $ where a, b and c are numbers

You can sketch quadratic function in 4 steps. I will explain these steps in following examples.

Example 1:

Sketch the graph of the quadratic function

$$ {\color{blue}{ f(x) = x^2+2x-3 }} $$

Solution:

In this case we have $ a=1, b=2 $ and $c=-3$

STEP 1: Find the vertex.

To find x - coordinate of the vertex we use formula:

$$ x=-\frac{b}{2a} $$

So, we substitute $1$ in for $a$ and $2$ in for $b$ to get

$$ x=-\frac{b}{2a} = -\frac{2}{2\cdot1} = -1 $$

To find y - coordinate plug in $x=-1$ into the original equation:

$$ y = f(-1) = (-1)^2 + 2\cdot(-1) - 3 = 1 - 2 - 3 = -4 $$

So, the vertex of the parabola is $ {\color{red}{ (-1,-4) }} $

STEP 2: Find the y-intercept.

To find y - intercept plug in $x=0$ into the original equation:

$$ f(0) = (0)^2 + 2\cdot(0) - 3 = 0 - 0 - 3 = -3 $$

So, the y-intercept of the parabola is $ {\color{blue}{ y = -3 }} $

STEP 3: Find the x-intercept.

To find x - intercept solve quadratic equation $f(x)=0$ in our case we have:

$$ x^2+2x-3 = 0 $$

Solutions for this equation are:

$$ {\color{blue}{ x_1 = -3 }} ~~~\text{and}~~~ {\color{blue}{ x_2 = 1 }} $$

( to learn how to solve quadratic equation use quadratic equation solver )

STEP 4: plot the parabola.

Example 2:

Sketch the graph of the quadratic function

$$ {\color{blue}{ f(x) = -x^2+2x-2 }} $$

Solution:

Here we have $ a=-1, b=2 $ and $c=-2$

The x-coordinate of the vertex is:

$$ {\color{blue}{ x = -\frac{b}{2a} }} = -\frac{2}{2\cdot(-1)}= 1 $$

The y-coordinate of the vertex is:

$$ y = f(1) = -1^2+2\cdot1-2 = -1 + 2 - 2 = -1 $$

The y-intercept is:

$$ y = f(0) = -0^2+2\cdot0-2 = -0 + 0 - 2 = -2 $$

In this case x-intercept doesn't exist since equation $-x^2+2x-2=0$ does not has the solutions (use quadratic equation solver to check ). So, in this case we will plot the graph using only two points