Section 7.2: Linear and Absolute Value Functions

7.2 Lecture Guide: Linear and Absolute Value Functions Objective: Use the slope-intercept form of a linear equation.

Linear Function Algebraically A function of the formis called a linear function. Graphical Example Verbally The graph of is a straight line. Each point on this line satisfies this equation. This line rises 2-units for each 1-unit move to the right.

Linear Function Algebraically A function of the formis called a linear function. Numerical Example Verbally In the table for, each 1-unit increase in x produces a 2-unit increase in y.

Use the slope and y-intercept to graph each line. 1. Slope: ______ y-intercept: ______ Graph:

Use the slope and y-intercept to graph each line. 2. Slope: ______ y-intercept: ______ Graph:

Use the given graph or table to determine the slope of the line, the y-intercept of the graph, and the equation of the line in slope-intercept form. 3. Graph: Slope: ______ y-intercept: ______ Equation: __________________

Use the given graph or table to determine the slope of the line, the y-intercept of the graph, and the equation of the line in slope-intercept form. 4. Table: Slope: ______ y-intercept: ______ Equation: __________________

Objective: Identify a function as an increasing or decreasing function.

Increasing and Decreasing Functions Graphical Example Verbally Increasing Function A function is increasing over its entire domain if the graph rises as it moves from left to right. For all x-values, as the x-values increase, the y-values also increase.

Increasing and Decreasing Functions Graphical Example Verbally Decreasing Function A function is decreasing over its entire domain if the graph drops as it moves from left to right. For all x-values, as the x-values increase, the y-values decrease.

5. Use the graph of each function to identify the function as an increasing function, a decreasing function, or a function that is neither increasing nor decreasing.

6.

7. Use the graph of each function to identify the function as an increasing function, a decreasing function, or a function that is neither increasing nor decreasing.

8. For a linear function in the form: (a) The function is increasing if __________________. (b) The function is decreasing if __________________. (c) The function is neither increasing nor decreasing if __________________.

Use the equation defining each function to identify the function as an increasing function, a decreasing function, or a function that is neither increasing nor decreasing. Use a graphing calculator only as a check. 9.

10. Use the equation defining each function to identify the function as an increasing function, a decreasing function, or a function that is neither increasing nor decreasing. Use a graphing calculator only as a check.

11. Use the equation defining each function to identify the function as an increasing function, a decreasing function, or a function that is neither increasing nor decreasing. Use a graphing calculator only as a check.

Objective: Determine where a function is positive or negative. Objective: Analyze the graph of an absolute value function.

Verbally GraphicallyAlgebraically Absolute Value Function Numerically This V-shaped graph opens upward with the vertex of the V-shape at The minimum y-value is 0. The domain, the projection of this graph onto the x-axis, is The range, the projection of this graph onto the y-axis, is The function is decreasing for and increasing for Vertex

Complete the table and graph each absolute value function. 12. Equation:

(a) Does the graph open up or down? (b) Determine the vertex. (c) Determine the maximum or minimum y-value. (d) Determine the domain. (e) Determine the range.

Complete the table and graph each absolute value function. 13. Equation:

(a) Does the graph open up or down? (b) Determine the vertex. (c) Determine the maximum or minimum y-value. (d) Determine the domain. (e) Determine the range.

Use the graph of each absolute value function to determine the interval of x-values for which the function is increasing and the interval of x-values for which the function is decreasing. 14. Increasing ____________ Decreasing ____________

Use the graph of each absolute value function to determine the interval of x-values for which the function is increasing and the interval of x-values for which the function is decreasing. 15. Increasing ____________ Decreasing ____________

Positive and Negative Functions Positive Function: A function is positive when the output value is positive. On the graph of this occurs at points above the x-axis. The y-values are all positive. Negative Function: A functionis negative when the output value is negative. On the graph ofthis occurs at points below the x-axis. The y-values are all negative.

16. (a) Determine the x-values for which the function is positive. (b) Determine the x-values for which the function is negative.

(a) Determine the x-values for which the function is positive. 17. (b) Determine the x-values for which the function is negative.

Complete the table, graph the absolute value function and determine the following. 18.

Complete the table, graph the absolute value function and determine the following. 18. (a) Vertex (b) The maximum or minimum y-value (c) Domain (d) Range

Complete the table, graph the absolute value function and determine the following. 18. (e) The x-values for which the function is positive (f) The x-values for which the function is negative

Complete the table, graph the absolute value function and determine the following. 18. (g) The x-values for which the function is increasing (h) The x-values for which the function is decreasing

Absolute Value Expression Verbally Graphically Equivalent Expression Solving Absolute Value Equations and Inequalities For any real numbers x and a and positive real number d: x is either d units left or right of a. a − d a a + d

Absolute Value Expression Verbally Graphically Equivalent Expression Solving Absolute Value Equations and Inequalities For any real numbers x and a and positive real number d: x is less than d units from a. ( ) a − d a a + d

Absolute Value Expression Verbally Graphically Equivalent Expression Solving Absolute Value Equations and Inequalities For any real numbers x and a and positive real number d: x is more than d units from a. a − d a a + d ( )

Similar statements can also be made about the order relations less than or equal to and greater than or equal to Expressions with d negative are examined in the group exercises at the end of this section.

Solve each equation and inequality. 19.

Solve each equation and inequality. 20.

Solve each equation and inequality. 21.

Solve each equation and inequality. 22.

23. Determine the profit and loss intervals for the profit function graphed below. The x-variable represents the number of units of production and y-variable represents the profit generated by the sale of this production. Profit Units Profit interval: Loss interval: