Section 3.1 – Quadratic Functions

Definition and General Form

A quadratic function is any function that can be written in the form:

• a, b, c are real numbers.

• a ≠ 0 (if a = 0, the function is linear, not quadratic).

The graph of a quadratic function is called a parabola.

Standard Form of a Quadratic Function

The standard form of a quadratic function is:

• The vertex of the parabola is at the point (h, k).

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

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

Completing the square can be used to rewrite a quadratic function from general form to standard form.

Graphing Parabolas in Standard Form

To graph a quadratic function in standard form, follow these steps:

1. Determine the direction: Check the sign of a to see if the parabola opens up or down.

2. Find the vertex: The vertex is at (h, k).

3. Find the x-intercepts: Solve f(x) = 0 for real solutions (these are the x-intercepts).

4. Find the y-intercept: Substitute x = 0 into the function to find f(0).

5. Plot points: Plot the vertex, intercepts, and additional points as needed. Connect with a smooth curve.

Example:

Graph using transformations.

• a = -1 (parabola opens downward).

• Vertex: (h, k) = (1, 4).

• Axis of symmetry: x = 1.

• y-intercept: .

• x-intercepts: Set : or .

• Minimum/Maximum: Since a < 0, the vertex is a maximum at (1, 4).

• Domain: All real numbers, .

• Range: .

Graphing Quadratic Functions in General Form

For , use the following steps:

1. Direction: If a > 0, opens up; if a < 0, opens down.

2. Vertex: The x-coordinate is . The y-coordinate is .

3. x-intercepts: Solve for x.

4. y-intercept: .

5. Plot and connect points.

Example:

Graph .

• a = 2 > 0 (parabola opens upward).

• Vertex: . . So vertex is .

• x-intercepts: or .

• y-intercept: .

• Minimum/Maximum: Since a > 0, vertex is a minimum.

• Domain: .

• Range: .

Key Properties of Quadratic Functions

• Vertex: The highest or lowest point on the graph, depending on the direction the parabola opens.

• Axis of symmetry: Vertical line through the vertex, equation or .

• Maximum/Minimum Value: The y-value of the vertex.

• Domain: All real numbers, .

• Range: If a > 0, ; if a < 0, .

Applications of Quadratic Functions

Projectile Motion Example

The height of an arrow, , in feet, can be modeled by:

• Maximum height and when it occurs:

  • Vertex x-coordinate: feet from release.

  • Maximum height: feet.

• Horizontal distance before hitting the ground:

  • Set : .

  • Use quadratic formula: .

  • Solving gives feet (rounded to nearest foot).

• Graph: The graph is a downward-opening parabola with vertex at (200, 205).

Optimization Example: Maximizing Area

Chris has 120 feet of fencing to enclose a rectangular garden. Find the dimensions that maximize the area.

• Let width = w, length = l. Perimeter: .

• Area: .

• Maximum area occurs at vertex: .

• So, , (a square).

• Maximum area: square feet.

Optimization Example: Minimizing Product

Among all pairs of numbers whose difference is 8, find the pair whose product is as small as possible.

• Let the numbers be and .

• Product: .

• Minimum occurs at vertex: .

• Numbers: and .

• Minimum product: .

Summary Table: Key Features of Quadratic Functions

Additional info: The above notes include expanded explanations, step-by-step examples, and a summary table for clarity and completeness.