Quadratic Functions: Graphing and Analysis

Quadratic Functions

Definition and Standard Form

A quadratic function is a polynomial function of degree 2, typically written in the form:

• Standard form: $f(x) = ax^2 + bx + c$, where $a \neq 0$

• The graph of a quadratic function is called a parabola.

Key Terms:

• Vertex: The highest or lowest point on the parabola.

• Axis of Symmetry: A vertical line that divides the parabola into two mirror images.

• Direction: If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

Graphing Quadratic Functions by Plotting Points

To graph a quadratic function, you can create a table of values by substituting values for $x$ and calculating the corresponding $y$ values.

• Plot the points $(x, f(x))$ on the coordinate plane.

• Connect the points with a smooth curve to form the parabola.

Example: For $f(x) = x^2$:

x	f(x)
-2	4
-1	1
0	0
1	1
2	4

The graph is a parabola opening upward with vertex at (0,0).

Domain and Range of Quadratic Functions

• Domain: All real numbers, $(-\infty, \infty)$

• Range: If the parabola opens upward, $[k, \infty)$; if downward, $(-\infty, k]$, where $k$ is the $y$-coordinate of the vertex.

Example: For $f(x) = x^2$, the range is $[0, \infty)$.

Graphing Techniques for Quadratic Functions

Vertex Form and Axis of Symmetry

The vertex form of a quadratic function is:

• $f(x) = a(x-h)^2 + k$

• The vertex is at $(h, k)$.

• The axis of symmetry is the vertical line $x = h$.

Example: For $f(x) = (x-4)^2$, the vertex is at (4,0), and the axis of symmetry is $x=4$.

Finding the Vertex from Standard Form

For $f(x) = ax^2 + bx + c$, the vertex $(h, k)$ can be found using:

• $h = -\frac{b}{2a}$

• $k = f(h)$

Example: For $f(x) = 2x^2 - 4x + 1$:

• $h = -\frac{-4}{2 \times 2} = 1$

• $k = f(1) = 2(1)^2 - 4(1) + 1 = -1$

• Vertex: (1, -1)

Graphing Steps for Quadratic Functions

1. Find the vertex $(h, k)$.

2. Find the axis of symmetry $x = h$.

3. Find the $y$-intercept by evaluating $f(0)$.

4. Find the $x$-intercepts by solving $f(x) = 0$ (if real solutions exist).

5. Plot additional points as needed for accuracy.

6. Draw the parabola, making sure it is symmetric about the axis of symmetry.

Intervals of Increase and Decrease

• If the parabola opens upward ($a > 0$):

  • Decreasing on $(-\infty, h)$

  • Increasing on $(h, \infty)$

• If the parabola opens downward ($a < 0$):

  • Increasing on $(-\infty, h)$

  • Decreasing on $(h, \infty)$

Practice: Analyzing and Graphing Quadratic Functions

Example 1: $f(x) = -x^2 + 2$

• Vertex: (0, 2)

• Opens downward

• Axis of symmetry: $x = 0$

• Domain: $(-\infty, \infty)$

• Range: $(-\infty, 2]$

Example 2: $f(x) = -2(x+3)^2 + 1$

• Vertex: (-3, 1)

• Opens downward

• Axis of symmetry: $x = -3$

• Domain: $(-\infty, \infty)$

• Range: $(-\infty, 1]$

Example 3: $f(x) = x^2 - 2x - 3$

• Find $x$-intercepts: Solve $x^2 - 2x - 3 = 0$

• Find $y$-intercept: $f(0) = -3$

• Vertex: $h = 1$, $k = f(1) = -4$

• Axis of symmetry: $x = 1$

Summary Table: Properties of Quadratic Functions

Form	Vertex	Axis of Symmetry	Direction	Domain	Range
$f(x) = a(x-h)^2 + k$	$(h, k)$	$x = h$	Up if $a > 0$, Down if $a < 0$	$(-\infty, \infty)$	Up: $[k, \infty)$ Down: $(-\infty, k]$
$f(x) = ax^2 + bx + c$	$(h, k)$, $h = -\frac{b}{2a}$, $k = f(h)$	$x = h$	Up if $a > 0$, Down if $a < 0$	$(-\infty, \infty)$	Up: $[k, \infty)$ Down: $(-\infty, k]$

Additional info:

• Some steps and explanations were expanded for clarity and completeness.

• Graph sketches referenced in the notes are described in text, as images cannot be rendered here.