Graphing Quadratic Equations Study Guide

Algebra I

Graphing Quadratic Equations

Big Picture

Quadratic functions involve a variable being squared. The graphs of quadratic functions are parabolas that are usually transformed versions of the basic equation $y = x^{2}$ . Graphing quadratic functions give us more information about the x-intercepts, which are also the solutions to quadratic equations. We can also know the number of solutions to a quadratic equation by graphing it.

Key Terms

Quadratic Function: A nonlinear function that can be written as $y = a x^{2} + b x + c$ , where $a \neq 0$ .

Parabola: The U-shaped graph formed by a quadratic function.

Vertex: The lowest or highest point on the parabola.

Axis of Symmetry: The vertical line that divides the parabola into two symmetric parts (mirror images).

x-Intercept: The point where a line crosses the x-axis.

Quadratic Equation: A nonlinear equation that can be written as $a x^{2} + b x + c = 0$ where $a \neq 0$ .

Root (of a polynomial): The value that makes the polynomial equal to $0$ .

Zero (of a function): The value that makes the function equal to $0$ .

Quadratic Functions

The standard form of a quadratic function: $y = a x^{2} + b x + c, a \neq 0$ .

When graphed, all quadratic functions are parabolas. The values of $a$ , $b$ , and $c$ affect how the parabola looks.

The value of a stretches or shrinks the graph
$a > 1$ makes the graph skinnier
$a < 1$ makes the graph fatter

If a is negative, the graph curves downward.
If a is positive, the vertex is at the bottom of the parabola and is the minimum value of the function.
If a is negative, the vertex is at the top of the parabola and is the maximum value of the function.

$c$ is a constant that shifts the parabola up or down.
Increasing $c$ moves the parabola up; decreasing $c$ moves the parabola down.
The point $(0, c)$ is on the parabola.

$b$ is a constant that shifts the parabola left and right. The axis of symmetry is the line $x = - \frac{b}{2 a}$ .
Increasing $b$ moves the parabola to the left; decreasing b moves the parabola to the right.

Algebra I

Graphing Quadratic Equations cont.

Graphing Quadratic Functions

The most straightforward way to graph quadratic functions is to make a table of values, plot them, then connect the points with a smooth curve.

Intercept Form

Another way to graph quadratic functions is to rewrite the function in the intercept form.

Intercept form: $y = a (x - p) (x - q)$ , where $p$ and $q$ are the x-intercepts.

The intercept form is just the factored form of the quadratic function.

Using the zero-product property, when $y = 0, x = p or x = q$ .
$(p, 0)$ and $(q, 0)$ are the x-intercepts.

The intercept form is useful because it tells us:

The x-intercepts
The axis of symmetry is halfway between the two intercepts, so the axis of symmetry is $x = \frac{p + q}{2}$
Knowing the axis of symmetry, we can find the vertex by plugging $x = \frac{p + q}{2}$ into the function
If $a > 0$ , the graph opens up; if $a < 0$ , the graph opens down

With these points, we can draw the parabola.

Knowing the x-intercepts and one other point on the parabola, we can also write the quadratic function for the parabola.

Solving Quadratic Equations by Graphing

Finding the roots or zeroes of a quadratic equation is the same as solving the quadratic equation; we want to find the values of $x$ that make the equation equal to $0$ .

This is the same as finding the x-intercepts.

A quadratic equation is the same as a quadratic function where $y = 0$ .

The x-intercepts can be found by factoring the equation (rewritten in the intercept form) or by graphing.

Looking at the graph tells us the number of solutions to the quadratic equation.

The equation has $2$ solutions if the graph has $2$ x-intercepts (see graph on left).
The equation has $1$ solution if the graph has only $1$ x-intercept (see graph in middle).
The equation has no solutions if the graph has no x-intercepts (see graph on right).

Algebra I

Graphing Quadratic Equations

Big Picture

Key Terms

Quadratic Functions

Algebra I

Graphing Quadratic Equations cont.

Graphing Quadratic Functions

Intercept Form

Solving Quadratic Equations by Graphing

Notes