# Section 3.1 - Quadratic Functions and Parabolas

Section Objectives

1. Find the vertex, intercepts, and symmetry axis of a parabola.
2. Write a quadratic function in vertex form.
3. Solve application problems involving quadratic functions and parabolas.

A quadratic function is a 2nd degree polynomial function. Every quadratic function can be written in the form $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are real numbers with $a \ne 0$.

The graph of a quadratic function is a smooth U-shaped curve called a parabola. The turning point at the tip of the U is called the vertex, and the graph is symmetric about the vertical line through the vertex.

Any quadratic function $f(x) = ax^2+bx+c$ can be written in the equivalent vertex form

$f(x) = a (x-h)^2 + k$

by using a process called completing the square.

• The vertex is at the point $(h,k)$, where $\displaystyle h=-\frac{b}{2a}$ and $k=f(h)$.
• The axis of symmetry is the vertical line $x=h$.
• The parabola opens upward if $a>0$ and downward if $a<0$ .
• The graph is wider than the graph of $y=x^2$ if $-1 \le a \le 1$ , as wide if $a=1$, and narrower if $|a|>1$.
• The $x$-intercepts of the graph can be found by solving $f(x)=0$ (i.e., finding the zeros of $f$). If a parabola has two distinct $x$-intercepts, the vertex is directly between them.
• The $y$-intercept is $(0,f(0))$.
• The quadratic function $f$ attains an extreme value (max or min) at the vertex of its graph.

#### Examples

• Consider the parabola described by $y=x^2+6x-7$. Find the $x$-intercepts, the coordinates of the vertex, and the axis of symmetry. Finally, write the equation in vertex form and sketch the graph.

• Sketch the graph of $f(x)=-2(x-3)^2+4$. Describe the features of the graph/function.

• A ball is thrown upward with a velocity of $48$ feet per second from the top of a $144$-foot building. What is the maximum height of the ball? How long until the ball hits the ground? $h(t) = -16t^2 +48t+144$

• A quadratic function has a leading coefficient of $3$ and zeros $x=6$ and $x=-2$. Write its equation in standard form and vertex form.