Book cover for Algebra and Trigonometry Real Mathematics, Real People

Algebra and Trigonometry Real Mathematics, Real People

Ron Larson

ISBN #9781305071735

7th Edition

6,909 Questions

Group icon
234,597 Students Helped

Homework Questions

Right arrow
Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Quadratic functions play a crucial role in modeling physical phenomena and optimization problems. By rewriting quadratic functions in standard form, one can easily identify key characteristics such as the vertex, axis of symmetry, and the direction in which the parabola opens. Mastery of graph transformations, completing the square, and applying these concepts in practical situations enhances problem-solving skills in both academic and real-world applications.

Learning Objectives

1

Analyze the graphs of quadratic functions and understand how changes in the coefficient a, and transformations affect the graph.

2

Write quadratic functions in standard (vertex) form and use the form to quickly identify key features such as the vertex, axis of symmetry, and direction of opening.

3

Determine minimum or maximum values of quadratic functions and apply these concepts to solve real-life optimization problems.

4

Sketch quadratic function graphs by performing rigid transformations (reflections, shifts, stretches, and shrinks) of the parent function f(x) = x².

Key Concepts

CONCEPT

DEFINITION

Quadratic Function

A polynomial function of degree 2 that can be written in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0.

Standard (Vertex) Form

A form of a quadratic function written as f(x) = a(x – h)² + k, where (h, k) is the vertex of the parabola and the axis of symmetry is x = h.

Parabola

The graph of a quadratic function. It opens upward when a > 0 (having a minimum vertex) and downward when a < 0 (having a maximum vertex).

Transformation

Changes applied to the parent function such as vertical stretches/shrinks, horizontal shifts, and reflections, which alter the graph's shape and position.

Completing the Square

A method used to convert a quadratic function from standard form to vertex form by adding and subtracting an appropriate constant.

Example Problems

Example 1

A polynomial function with degree $n$ and leading coefficient $a_{n}$ is a function of the form $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{2} x^{2}+a_{1} x+a_{0}, a_{n} \neq 0,$ where $n$ is _____ and $a_{n}, a_{n-1}, \ldots, a_{2}, a_{1}, a_{0}$ are _____ members.

Example 2

A _____ function is a second-degree polynomial function, and its graph is called a _____.

Example 3

Is the quadratic function $f(x)=(x-2)^{2}+3$ written in standard form? Identify the vertex of the graph of $f$

Example 4

Does the graph of the quadratic function $f(x)=-3 x^{2}+5 x+2$ have a relative minimum value at its vertex?

Example 5

Match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), and (d).] $f(x)=(x-2)^{2}$
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

Describe the steps to rewrite f(x) = 2x² + 8x + 7 in standard form and identify the vertex.

STEP-BY-STEP ANSWER:

Step 1: Factor out the coefficient of x² from the x terms: f(x) = 2(x² + 4x) + 7.
Step 2: To complete the square, add and subtract (4/2)² = 4 inside the parentheses: f(x) = 2(x² + 4x + 4 - 4) + 7.
Step 3: Rewrite the quadratic expression as a perfect square: f(x) = 2[(x + 2)² - 4] + 7.
Step 4: Distribute and simplify: f(x) = 2(x + 2)² - 8 + 7, hence f(x) = 2(x + 2)² - 1.
Step 5: Identify the vertex from the standard form: Vertex = (-2, -1).
Final Answer: The quadratic function in standard form is f(x) = 2(x + 2)² - 1 with vertex (-2, -1).

Finding the Vertex by Completing the Square

QUESTION

Write the standard form of the parabola with vertex (1, 2) that passes through the point (3, -6).

STEP-BY-STEP ANSWER:

Step 1: Start with the vertex form: f(x) = a(x - 1)² + 2.
Step 2: Substitute the point (3, -6) into the equation: -6 = a(3 - 1)² + 2.
Step 3: Simplify the equation: -6 = 4a + 2.
Step 4: Solve for the coefficient a: 4a = -8, therefore a = -2.
Final Answer: The standard form of the parabola is f(x) = -2(x - 1)² + 2.

Writing the Equation of a Parabola

Scroll left
Scroll right

Common Mistakes

  • Incorrectly factoring out the coefficient when completing the square.
  • Sign errors during reflection and translation steps.
  • Misidentifying the direction in which a parabola opens due to neglecting the sign of the leading coefficient.
  • Forgetting to correctly compute and interpret the vertex, leading to an incorrect graph sketch.
  • Overlooking the impact of vertical stretches and shrinks when comparing transformed quadratic functions with the parent function.