Analyzing Graphs of Quadratic Functions (Section 3.3)

Quadratic Functions and Their Graphs

Introduction to Quadratic Functions

Quadratic functions are a fundamental class of polynomial functions, typically written in the form $f(x) = ax^2 + bx + c$, where $a \neq 0$. The graph of a quadratic function is called a parabola, which can open upwards or downwards depending on the sign of $a$.

• Vertex: The point $(h, k)$ where the parabola turns; it represents the maximum or minimum value of the function.

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

• Maximum/Minimum Value: The $y$-coordinate of the vertex; if $a > 0$, the vertex is a minimum, if $a < 0$, it is a maximum.

Standard and Vertex Forms of Quadratic Functions

Quadratic functions can be expressed in two main forms:

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex Form: $f(x) = a(x - h)^2 + k$

Converting from standard to vertex form often involves completing the square.

Graphing Quadratic Functions

To graph a quadratic function, identify the vertex, axis of symmetry, and direction of opening. The vertex form $f(x) = a(x-h)^2 + k$ makes these features explicit:

• Vertex: $(h, k)$

• Axis of Symmetry: $x = h$

• Opens Upward: if $a > 0$

• Opens Downward: if $a < 0$

Example graphs show how changes in $h$ and $k$ shift the parabola horizontally and vertically.

Finding the Vertex by Completing the Square

Completing the square is a method to rewrite $f(x) = ax^2 + bx + c$ in vertex form:

• Group $x^2$ and $x$ terms.

• Add and subtract the necessary constant to complete the square.

• Rewrite as $a(x-h)^2 + k$.

Example: Find the vertex, axis of symmetry, and minimum value of $f(x) = x^2 + 10x + 23$.

• Complete the square: $f(x) = x^2 + 10x + 23$ $= x^2 + 10x + 25 - 25 + 23$ $= (x + 5)^2 - 2$

• Vertex: $(-5, -2)$

• Axis of symmetry: $x = -5$

• Minimum value: $-2$

Table of Values for Graphing

Constructing a table of $x$ and $y$ values helps plot the parabola:

x	y
-5	-2
-6	-1
-4	-1
-7	2
-3	2

Another Example: Completing the Square

Example: Find the vertex, axis of symmetry, and minimum value of $g(x) = \frac{1}{2}x^2 - 4x + 8$.

• Complete the square: $g(x) = \frac{1}{2}(x^2 - 8x) + 8$ $= \frac{1}{2}(x^2 - 8x + 16 - 16) + 8$ $= \frac{1}{2}(x^2 - 8x + 16) - 8 + 8$ $= \frac{1}{2}(x - 4)^2$

• Vertex: $(4, 0)$

• Axis of symmetry: $x = 4$

• Minimum value: $0$

The graph is a vertical stretching of $y = x^2$, shifted 4 units to the right.

Vertex Formula for Quadratic Functions

The vertex of $f(x) = ax^2 + bx + c$ can be found using:

• $x$-coordinate: $-\frac{b}{2a}$

• $y$-coordinate: $f\left(-\frac{b}{2a}\right)$

Thus, the vertex is: $\left(-\frac{b}{2a},\ f\left(-\frac{b}{2a}\right)\right)$

Analyzing Quadratic Functions: Maximum, Minimum, Range, and Intervals

Given $f(x) = -x^2 + 14x - 47$:

• Find the vertex: $x = -\frac{b}{2a} = -\frac{14}{2(-1)} = 7$ $f(7) = -(7)^2 + 14 \cdot 7 - 47 = 2$ Vertex: $(7, 2)$

• Since $a < 0$, the parabola opens downward; the vertex is a maximum.

• Maximum value: $2$

• Range: $(-\infty, 2]$

• Increasing on $(-\infty, 7)$; decreasing on $(7, \infty)$

Applied Problem: Maximizing Area

Quadratic functions are often used to solve optimization problems.

• Example: A landscaper has 24 ft of stone wall to enclose a rectangular pond next to a garden wall (which forms one side).

• Let $w$ be the width perpendicular to the wall; length is $24 - 2w$.

• Area function: $A(w) = (24 - 2w)w = -2w^2 + 24w$

• Maximum area occurs at the vertex: $w = -\frac{24}{2(-2)} = 6$ Length: $24 - 2 \cdot 6 = 12$ ft Maximum area: $12 \cdot 6 = 72$ ft$^2$

Quadratic modeling is essential for solving real-world maximum and minimum problems.

Summary Table: Features of Quadratic Functions

Feature	Description
Vertex	$(h, k)$ in vertex form; $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$ in standard form
Axis of Symmetry	$x = h$ in vertex form; $x = -\frac{b}{2a}$ in standard form
Maximum/Minimum	$k$ in vertex form; $f\left(-\frac{b}{2a}\right)$ in standard form
Direction	Upward if $a > 0$, downward if $a < 0$

Additional info: These notes expand on the provided examples and definitions, adding context for the use of quadratic functions in optimization and graph analysis, as well as summarizing key formulas and properties for Precalculus students.