Algebra 3.1 notes

College Algebra - Chapter 3: Polynomial and Rational Functions

Section 3.1: Quadratic Functions

• Objectives:

  • Recognize characteristics of parabolas.

  • Graph parabolas.

  • Determine a quadratic function’s minimum or maximum value.

  • Solve problems involving a quadratic function’s minimum or maximum value.

Standard Form of a Quadratic Function

• The quadratic function can be expressed in vertex form:

  • General form: ( f(x) = a(x - h)^2 + k )

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

  • The parabola is symmetric about the line ( x = h ).

  • Direction of Opening:

    • If ( a > 0 ), opens upward.

    • If ( a < 0 ), opens downward.

Graphing Quadratic Functions in Vertex Form

1. Determine Direction:

   • Check the value of ( a ).

2. Find the Vertex:

   • Coordinate is (h, k).

3. Find x-intercepts:

   • Solve ( f(x) = 0 ) for x-intercepts.

4. Find the y-intercept:

   • Calculate ( f(0) ).

5. Plot Points:

   • Plot intercepts and vertex, and draw a smooth curve.

Example 1: Graphing Quadratic Function

• Step 1: The parabola opens downward (( a < 0 )).

• Step 2: The vertex is at (1, 4).

• Step 3: x-intercepts found to be (3, 0).

• Step 4: y-intercept is (0, 3).

• Graph Summary:

  • Opens downward; axis of symmetry: x = 1; intercepts: x = 3, y = 3.

The Vertex of a Parabola

For the quadratic function ( f(x) = ax^2 + bx + c ):

• Vertex Coordinates:

  • x-coordinate: ( x = -\frac{b}{2a} )

  • y-coordinate calculated by evaluating ( f ) at the x-coordinate.

Graphing Quadratic Functions in Standard Form

• Steps to Graph:

  1. Determine the direction (upward if ( a > 0 ), downward if ( a < 0 )).

  2. Calculate the vertex using the formula for x and substituting back to find y.

  3. Solve for x-intercepts (roots).

  4. Find the y-intercept at (0, c).

  5. Plot all points and connect with a smooth curve.

Example 3: Graphing Quadratic Function in Standard Form

• Steps:

  1. The parabola opens downward.

  2. The vertex calculated to be (2, 5).

  3. x-intercepts identified.

  4. y-intercept identified as (0, 1).

  5. Complete the graph with intercepts and vertex points.

Minimizing and Maximizing Quadratic Functions

• General Criterion:

  • If ( a > 0 ): Minimum at vertex.

  • If ( a < 0 ): Maximum at vertex.

• Determine the vertex to find the minimum or maximum value of the quadratic function.

Example 4: Analyzing Quadratic Function Without Graphing

1. Evaluate: Given ( a = 4 ) implies a minimum value exists.

2. Locate the vertex and evaluate the function for maximum/minimum values.

3. Identify domain (all real numbers) and range, based on vertex location and direction of parabola.

Application: Strategy for Solving Optimization Problems

• Steps to Solve:

  1. Identify the quantity to optimize (maximize/minimize).

  2. Express this quantity as a function in one variable.

  3. Write the function in the standard quadratic form.

  4. Determine the vertex based on a or b to find extremum.

  5. State the solution clearly.

Example 7: Maximizing Area Using Quadratic Functions

1. Define the problem: 120 feet of fencing must enclose an area.

2. Express Area: Area as a function of dimensions using given perimeter constraints.

3. Calculate Extremum: Identify maxima/minima based on function parameters derived from dimensions.

4. Conclude dimensions for maximum area: Dimensions derived yield 30 ft x 30 ft as optimal; maximum area calculated.