Analyzing Graphs of Quadratic Functions: Vertex, Axis of Symmetry, and Extrema

Quadratic Functions and Their Graphs

Introduction to Quadratic Functions

Quadratic functions are fundamental in precalculus and are represented by equations of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. The graph of a quadratic function is a parabola, which can open upward or downward depending on the sign of $a$.

• Vertex: The point $(h, k)$ where the parabola changes direction.

• Axis of Symmetry: The vertical line $x = h$ that passes through the vertex and divides the parabola into two symmetric halves.

• Maximum/Minimum Value: The highest or lowest point on the graph, corresponding to the $y$-value of the vertex.

Graphing Quadratic Functions in Vertex Form

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by:

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

• Vertex: $(h, k)$

• Axis of Symmetry: $x = h$

• Direction: If $a > 0$, the parabola opens upward (minimum value). If $a < 0$, it opens downward (maximum value).

Example 1

Given $f(x) = 2(x + 3)^2 - 2$, the vertex is $(-3, -2)$ and the axis of symmetry is $x = -3$. The minimum value is $-2$.

Example 2

Given $f(x) = -2(x - 1)^2 + 3$, the vertex is $(1, 3)$ and the axis of symmetry is $x = 1$. The maximum value is $3$.

Finding the Vertex, Axis of Symmetry, and Extrema from Standard Form

Standard Form and Completing the Square

Quadratic functions can also be written in standard form: $f(x) = ax^2 + bx + c$. To find the vertex, we use:

• x-coordinate of vertex: $x = -\frac{b}{2a}$

• y-coordinate of vertex: Substitute $x$ into $f(x)$

Example

For $f(x) = x^2 + 10x + 23$:

• $a = 1$, $b = 10$, $c = 23$

• $x = -\frac{10}{2 \times 1} = -5$

• $y = (-5)^2 + 10(-5) + 23 = 25 - 50 + 23 = -2$

• Vertex: $(-5, -2)$

• Axis of Symmetry: $x = -5$

• Minimum Value: $-2$

x	y
-7	2
-6	2
-5	-2
-4	2
-3	2

Maximum and Minimum Values

Determining Extrema

The maximum or minimum value of $f(x)$ occurs at the vertex. The nature of the extremum depends on the sign of $a$:

• If $a > 0$: Parabola opens upward; vertex is a minimum.

• If $a < 0$: Parabola opens downward; vertex is a maximum.

Applied Problems Involving Quadratic Functions

Projectile Motion Example

Quadratic functions model real-world phenomena such as projectile motion. For example, the height $s(t)$ of a stone thrown upward is given by:

$s(t) = -16t^2 + 60t + 30$

• Maximum Height: Occurs at $t = -\frac{b}{2a}$

• Application: Substitute $t$ into $s(t)$ to find the maximum height.

Area Optimization Example

Quadratic functions can be used to maximize area, such as finding the dimensions of a rectangular pond with a fixed perimeter.

• Set up a quadratic equation for area in terms of one variable.

• Find the vertex to determine the maximum area.

Summary Table: Properties of Quadratic Functions

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

Key Formulas

• Vertex (Standard Form): $x = -\frac{b}{2a}$

• Vertex Form: $f(x) = a(x-h)^2 + k$

• Axis of Symmetry: $x = h$ (vertex form), $x = -\frac{b}{2a}$ (standard form)

Additional info:

• TI-84 or similar calculators can be used to graph quadratic functions and find vertex form efficiently.

• Quadratic functions are essential for modeling optimization and motion problems in mathematics and science.