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.

**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.

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.

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.

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).