Quadratic Functions: Graphs, Vertices, and Applications

Quadratic Functions and Their Graphs

Introduction to Quadratic Functions

Quadratic functions are a fundamental topic in precalculus, characterized by equations of the form f(x) = ax^2 + bx + c, where a ≠ 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: A vertical line x = h that divides the parabola into two mirror-image halves.

• Maximum/Minimum Value: The y-coordinate of the vertex; maximum if the parabola opens down, minimum if it opens up.

Standard and Vertex Forms

Quadratic functions can be written in different forms to facilitate graphing and analysis.

• 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.

Analyzing Graphs of Quadratic Functions

Completing the Square

Completing the square is a method used to rewrite a quadratic function in vertex form, making it easier to identify the vertex and axis of symmetry.

• Given $f(x) = ax^2 + bx + c$, factor out a if necessary, then complete the square for the quadratic and linear terms.

• Example: $f(x) = x^2 + 10x + 23$

Steps:

• $f(x) = x^2 + 10x + 23$

• $= x^2 + 10x + 25 - 25 + 23$

• $= (x + 5)^2 - 2$

Thus, the vertex is $(-5, -2)$ and the axis of symmetry is $x = -5$.

Graphing Quadratic Functions

To graph a quadratic function, identify the vertex, axis of symmetry, and plot several points on either side of the vertex.

• For $f(x) = x^2 + 10x + 23$:

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

The vertex is at $(-5, -2)$, and the axis of symmetry is $x = -5$.

Example: Vertex and Axis of Symmetry

Given $g(x) = \frac{x^2}{2} - 4x + 8$, complete the square:

• $g(x) = \frac{1}{2}(x^2 - 8x) + 8$

• $= \frac{1}{2}(x^2 - 8x + 16 - 16) + 8$

• $= \frac{1}{2}(x - 4)^2$

Vertex: $(4, 0)$; Axis of symmetry: $x = 4$; Minimum value: $0$.

Vertex of a Parabola: Formula

Finding the Vertex

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

• x-coordinate: $x = -\frac{b}{2a}$

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

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

Applications: Maximum and Minimum Problems

Applied Example: Maximizing Area

Quadratic functions are often used to solve real-world problems involving maximum or minimum values, such as maximizing the area of a rectangle with a fixed perimeter.

• Problem: A landscaper wants to enclose a rectangular koi pond next to a garden wall, using 24 ft of stone wall for three sides.

• Let w = width of the pond.

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

To find the maximum area, locate the vertex:

• $w = -\frac{24}{2 \times -2} = 6$ ft

• Length: $24 - 2w = 12$ ft

• Maximum area: $12 \times 6 = 72$ ft2

Thus, the maximum possible area is 72 ft2 when the pond is 6 ft wide and 12 ft long.

Summary Table: Key Properties of Quadratic Functions

Form	Vertex	Axis of Symmetry	Maximum/Minimum Value	Direction
$f(x) = a(x-h)^2 + k$	$(h, k)$	$x = h$	$k$	Up if $a > 0$, Down if $a < 0$
$f(x) = ax^2 + bx + c$	$\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$	$x = -\frac{b}{2a}$	$f\left(-\frac{b}{2a}\right)$	Up if $a > 0$, Down if $a < 0$

Intervals of Increase and Decrease

Behavior of Quadratic Functions

The function increases on the interval to the left of the vertex and decreases to the right (if the parabola opens down), or vice versa (if it opens up).

• Increasing: $(-\infty, h)$ for $a > 0$; $(h, \infty)$ for $a < 0$

• Decreasing: $(h, \infty)$ for $a > 0$; $(-\infty, h)$ for $a < 0$

Where $h$ is the x-coordinate of the vertex.

Conclusion

Quadratic functions are essential in precalculus, providing tools for graphing, analyzing, and solving real-world problems involving maximum and minimum values. Mastery of completing the square, vertex identification, and graph interpretation is crucial for success in further mathematical studies.