## 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 $$y = ax^2 + bx + c$$, where $$a ≠ 0$$.

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 $$ax^2 + bx + c = 0$$ where $$a ≠ 0$$.

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 $$0$$.

The standard form of a quadratic function: $$y = ax^2 + bx + c, a ≠ 0$$.

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 $$x = -\frac{b}{2a}$$.
• Increasing $$b$$ moves the parabola to the left; decreasing b moves the parabola to the right.

## Algebra I

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 $$y = 0, x = p \text{ or } x = q$$.
• $$(p, 0)$$ 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).