Lesson 89: Identifying Characteristics of Quadratic Functions

New Concept

The axis of symmetry for the graph of a quadratic equation $y = ax^2 + bx + c$ is $x = -\frac{b}{2a}$.

What’s next

Next, you’ll apply this formula and other tools to analyze the key features of parabolas, like their vertex and zeros.

Property

The vertex of a parabola is the highest or lowest point on a parabola. It is the parabola's 'turning point.' The minimum or maximum is the y-coordinate of the vertex.

Explanation

Imagine a roller coaster track shaped like a 'U'. The vertex is the very bottom of the dip (a minimum) or the top of the hill (a maximum). It's the point where the ride changes direction, telling you the absolute lowest or highest the coaster goes on its path!

Examples

• For the parabola $y=x^2-6$, the vertex at $(0, -6)$ is the lowest point, so the minimum value is $-6$.
• For the parabola $y=-x^2+2x+7$, the vertex at $(1, 8)$ is the highest point, so the maximum value is $8$.

Property

A zero of a function is the value of $x$ that makes $f(x) = 0$. The zeros are the same as the x-intercepts because $y = 0$ on the x-axis. These are also known as the function's roots.

Explanation

Think of zeros as the 'touchdown' points where the parabola hits the ground (the x-axis). At these spots, the function's value is exactly zero. Finding these roots is like solving a puzzle to discover where your graph crosses the horizontal line, a key clue to its location.

Examples

• The function $f(x) = -x^2 + 2x + 3$ has zeros at $x=-1$ and $x=3$, where the graph crosses the x-axis.
• The function $f(x) = 3x^2 + 12x + 12$ has one zero at $x=-2$, where the vertex touches the x-axis.
• The function $f(x) = x^2 - 4x + 6$ has no real zeros because its graph never crosses the x-axis.

A parabola's symmetry is perfectly balanced between its roots. Let's find that balance point using the key idea of finding the axis of symmetry from the zeros of a function.

Example Problem

Find the axis of symmetry for a parabola with zeros at $(-2, 0)$ and $(8, 0)$.

Step-by-step

1. The axis of symmetry is a vertical line that passes through the vertex, exactly halfway between the zeros. We can find its x-coordinate by averaging the x-coordinates of the zeros.
2. Average the zeros to find the x-coordinate of the vertex:

3. The x-coordinate of the vertex is $3$.
4. Since the axis of symmetry is the vertical line passing through the vertex, its equation is $x = 3$.

Property

The axis of symmetry for the graph of a quadratic equation $y = ax^2 + bx + c$ is $x = -\frac{b}{2a}$.

Explanation

This formula is your secret weapon for finding the invisible line that splits a parabola into two perfect mirror images. This line always passes straight through the vertex, so the formula also gives you the x-coordinate of the vertex! Just plug in the 'a' and 'b' from your equation.

Examples

• For $y = x^2 + 6x + 5$, the axis of symmetry is $x = -\frac{6}{2(1)} = -3$.
• For $y = -2x^2 + 8x - 1$, the axis of symmetry is $x = -\frac{8}{2(-2)} = 2$.
• For a golf ball's height $y = -16t^2 + 160t + 10$, the time to reach the max height is $t = -\frac{160}{2(-16)} = 5$ seconds.