Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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Concept explainers

Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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solve for a. and b.

### Educational Content on Linear Functions #### The Graph of a Linear Function The graph of a linear function \( f \) is shown below. ##### Tasks: 1. **Identify the slope, y-intercept, and x-intercept.** 2. **Write the equation that defines \( f \).** **Identifying Key Features:** - The slope is \( \boxed{-1} \). *(Type an integer or a simplified fraction.)* **Graph Explanation:** The graph displayed is a Cartesian coordinate plane with both x and y axes ranging from -10 to 10. The linear function graph is a straight line that intersects the y-axis at \( y = 8 \) and the x-axis at \( x = 8 \). **Steps to Identify:** 1. **Slope (m):** - The slope of a line is calculated using the formula \( \frac{\Delta y}{\Delta x} \). - On the graph, the line goes down by 8 units vertically (change in \( y \)) for every 8 units it moves horizontally (change in \( x \)), giving a slope of \( \frac{-8}{8} = -1 \). 2. **Y-intercept (b):** - The y-intercept is the point where the line crosses the y-axis. On this graph, the y-intercept is at \( y = 8 \). 3. **X-intercept:** - The x-intercept is the point where the line crosses the x-axis. On this graph, the x-intercept is at \( x = 8 \). **Writing the Equation of the Line:** - For a linear function in the form \( y = mx + b \): - Given the slope \( m = -1 \) and the y-intercept \( b = 8 \), - The equation defining \( f \) is \( y = -x + 8 \). By understanding these key elements, students can better grasp the properties and equations of linear functions.
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Transcribed Image Text:### Educational Content on Linear Functions #### The Graph of a Linear Function The graph of a linear function \( f \) is shown below. ##### Tasks: 1. **Identify the slope, y-intercept, and x-intercept.** 2. **Write the equation that defines \( f \).** **Identifying Key Features:** - The slope is \( \boxed{-1} \). *(Type an integer or a simplified fraction.)* **Graph Explanation:** The graph displayed is a Cartesian coordinate plane with both x and y axes ranging from -10 to 10. The linear function graph is a straight line that intersects the y-axis at \( y = 8 \) and the x-axis at \( x = 8 \). **Steps to Identify:** 1. **Slope (m):** - The slope of a line is calculated using the formula \( \frac{\Delta y}{\Delta x} \). - On the graph, the line goes down by 8 units vertically (change in \( y \)) for every 8 units it moves horizontally (change in \( x \)), giving a slope of \( \frac{-8}{8} = -1 \). 2. **Y-intercept (b):** - The y-intercept is the point where the line crosses the y-axis. On this graph, the y-intercept is at \( y = 8 \). 3. **X-intercept:** - The x-intercept is the point where the line crosses the x-axis. On this graph, the x-intercept is at \( x = 8 \). **Writing the Equation of the Line:** - For a linear function in the form \( y = mx + b \): - Given the slope \( m = -1 \) and the y-intercept \( b = 8 \), - The equation defining \( f \) is \( y = -x + 8 \). By understanding these key elements, students can better grasp the properties and equations of linear functions.
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