Quadratic Functions: Graphs, Properties, and Applications

Quadratic Functions and Their Graphs

Understanding Quadratic Functions

Quadratic functions are a fundamental class of functions in algebra, characterized by the general form $y = ax^2 + bx + c$, where $a \neq 0$. Their graphs are called parabolas, which can open either upward or downward depending on the sign of the leading coefficient $a$.

• Standard Form: $y = ax^2 + bx + c$

• Vertex Form: $y = a(x-h)^2 + k$

• Axis of Symmetry: The vertical line that passes through the vertex, given by $x = -\frac{b}{2a}$ in standard form or $x = h$ in vertex form.

• Vertex: The highest or lowest point on the parabola, depending on whether it opens downward or upward, respectively.

Key Terminology and Equivalences

Several terms are used interchangeably when discussing quadratic functions:

• Zeros of a function: The x-values where $f(x) = 0$.

• X-intercepts: The points where the graph crosses the x-axis.

• Solutions to the equation: The values of $x$ that satisfy $ax^2 + bx + c = 0$.

Properties of Parabolas

Opening Direction and Vertex Type

The direction in which a parabola opens is determined by the sign of $a$:

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• If $a > 0$, the parabola opens upward and the vertex is a minimum.

• If $a < 0$, the parabola opens downward and the vertex is a maximum.

Finding the Vertex

The vertex $(h, k)$ of a parabola in standard form can be found using:

• x-coordinate: $x = -\frac{b}{2a}$

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

The axis of symmetry is the vertical line $x = -\frac{b}{2a}$.

Vertex Form of a Parabola

Writing a quadratic in vertex form makes it easy to identify the vertex:

• Vertex Form: $y = a(x-h)^2 + k$

• The vertex is at $(h, k)$.

• The axis of symmetry is $x = h$.

Graphing Quadratic Functions

Step-by-Step Graphing

1. Identify the vertex using the formulas above.

2. Find additional points by substituting x-values into the function.

3. Plot the vertex and points on the coordinate plane.

4. Draw the axis of symmetry through the vertex.

5. Sketch the parabola opening up or down as determined by $a$.

Example: Graphing $y = x^2 + 12x + 32$

• Axis of symmetry: $x = -\frac{12}{2(1)} = -6$

• Vertex: $(-6, -4)$

• Plot additional points and sketch the parabola.

Solving Quadratic Equations

Determining Quadratic Nature and Vertex Type

To determine if a function is quadratic, check if the highest exponent of $x$ is 2. The sign of $a$ tells you if the graph is concave up or down, and whether the vertex is a maximum or minimum.

Estimating X-Intercepts from a Graph

The x-intercepts (or zeros) of a quadratic function are the points where the graph crosses the x-axis. These can be estimated visually from the graph or calculated algebraically by solving $ax^2 + bx + c = 0$.

Applications and Calculator Use

Using a Graphing Calculator to Find a Quadratic Model

To fit a quadratic model to data using a graphing calculator:

1. Enter x-values into L1 and y-values into L2 using the STAT → EDIT feature.

2. Select STAT → CALC → 5:QuadReg.

3. Press ENTER to compute the quadratic regression equation.

4. Construct the equation from the output values of $a$, $b$, and $c$.

Application Example: Maximizing Profit

Quadratic functions are often used in business to model profit, where the vertex represents the maximum profit point. For example, if $P(x) = -x^2 + 212x - 5500$, the maximum profit occurs at $x = -\frac{b}{2a}$ units sold.

• Find $x$ for maximum profit: $x = -\frac{212}{2(-1)} = 106$

• Calculate maximum profit: $P(106) = -(106)^2 + 212 \times 106 - 5500 = 5736$

Summary Table: Forms and Properties of Quadratic Functions

Form	Equation	Vertex	Axis of Symmetry	Opens
Standard	$y = ax^2 + bx + c$	$\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$	$x = -\frac{b}{2a}$	Up if $a > 0$, Down if $a < 0$
Vertex	$y = a(x-h)^2 + k$	$(h, k)$	$x = h$	Up if $a > 0$, Down if $a < 0$

Additional Resources

• For further study, refer to helpful videos and multimedia resources in your course platform.