AP Precalculus Section 1.8 Example: Find Where the Rational Function is Positive

Random AP Precalculus Problems (I found on the Internet). These are not official AP Collegeboard examples, but they will definitely get the job done!

If both the numerator and denominator of a rational function are polynomials of degree 1 (linear functions), determining the intervals where the rational function is positive or negative is relatively straightforward. Here's how you can do it:

1. Identify the critical point: Find the value of x where the numerator and denominator are equal to zero. Set the numerator equal to zero and solve for x, then set the denominator equal to zero and solve for x. These are your critical points.

2. Create a sign chart: On a number line, place the critical points you found in step 1. These points divide the number line into intervals.

3. Test a point in each interval: Choose a test point in each interval between the critical points and plug it into the original rational function.

4. Determine the sign: Evaluate the rational function at each test point. If the result is positive, the function is positive in that interval. If the result is negative, the function is negative in that interval.

5. Repeat for all intervals: Continue testing points in each interval until you've determined the sign of the function for all intervals.

By following these steps, you can identify the intervals in which the rational function is positive or negative. It's essential to understand that the linear nature of both the numerator and denominator simplifies this process, as the function only changes sign at the critical points.

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

/ nickperich

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa . #math #shorts #funny #help #onlineclasses #onlinelearning #online #study