Activity Depreciation Or What Happened to the Value of My New Car

5-Minute Check on Activity 2-2 Click the mouse button or press the Space Bar to display the answers. 1.What is the formula for slope? 2.Write the slopes for each of the following graphs and match up the equation of the line corresponding to the graphs. (y = x, y = -x, y = 4, x = 4)  y y 2 – y 1 Slope = m = =  x x 2 – x 1 m = 1 y x y x y x y x m = -1m = 0m = undefined y = xy = -xy = 4x = 4

Objectives Identify whether a situation can be modeled by a linear function Determine x and y intercepts of a graph Identify the practical meaning of x and y intercepts Develop the slope-intercept model of an equation of a line Use the slope-intercept formula to determine x and y intercepts Determine the zeros of a function

Vocabulary Vertical Intercept– is the point the graph crosses the vertical axis (dependent variable) Y-Intercept – is the point the graph crosses the y-axis Slope-Intercept Form – a line in the form of y = b + mx or y = mx + b (where b is the y-intercept) Horizontal Intercept– is the point the graph crosses the horizontal axis (independent variable) X-Intercept – is the point the graph crosses the x-axis. It is also known as a zero of the function Positive slope – m > 0, graph is increasing left to right Negative slope – m < 0, graph is decreasing left to right

Activity You have decided to by a new Honda Accord LX, but you are concerned about the value of the car depreciating over time. You search the Internet and obtain the following information from Accord LX –Suggested Retail Price $20,025 –Depreciation per Year $1,385 Complete the following table: Years of ownership Value in Dollars 20,

Activity Continued Questions: Is the value of the car a function of years of ownership? What is the independent and dependent variable? Select two ordered pairs and determine the average rate of change What are the units for the average rate of change? Years of ownership Value in Dollars20,02518,64017,25515,87013,1008,945 Yes Independent – years of ownership dependent - value average rate of change is for both dollars / year of ownership

Activity Continued Questions cont: Select another two ordered pairs and determine the average rate of change Is this a linear function? Why or why not? Is this function increasing, decreasing or constant? Years of ownership Value in Dollars20,02518,64017,25515,87013,1008,945 average rate of change is for both average rate of change is constant so it is linear decreasing (slope is negative)

Slope The slope is the average rate of change The slope is denoted by the letter m Geometry teachers figure slope using rise over run because it looks like triangles (the love of geometry) Algebraic formula is m = (y 2 – y 1 ) / (x 2 – x 1 )

Vertical Intercept The vertical intercept is the point where the graph crosses or intercepts the vertical axis – the dependent variable (usually y). It is often called the y-intercept. The corresponding x-value of the y- intercept is always zero. In practical terms the y-intercept represents one of the following: a) the value at time t = 0 (like our new car), the initial value b) the value of the function when x is zero Sometimes it has no practical meaning as we will see in later chapters.

Activity Continued Questions cont: What is the vertical intercept, value-intercept in our problem? What is the practical meaning of the v-intercept? What is the slope-intercept form of the line? Years of ownership Value in Dollars20,02518,64017,25515,87013,1008,945 20,025 (value of the vehicle ) the initial purchase price of the vehicle value = 20,025 – 1385x

Slope Intercept Form A line in slope-intercept form: y = mx + b where m represents the slope of the line (the average rate of change) and b represents the y-intercept (0, b) ∆y slope, m = ---- ∆x y 2 – y 1 rise = = = average rate of change x 2 – x 1 run

Example 1 Identify the slope and y-intercept of the line whose equation is given. Write the y-intercept as an ordered pair. a. y = -2x + 5 b. s = ⅜t + 2 c. q = 2 – r d. y = 5/6 + x/3 slope = -2 y-intercept: (0,5) slope = 3/8 y-intercept: (0,2) slope = -1 y-intercept: (0,2) slope = 1/3 y-intercept: (0,5/6)

Horizontal Intercept The point at which the graph meets or crosses the horizontal (or x) axis. This is usually called the x- intercept. The value of the independent variable (usually y) is zero. Also known as the “zeros” of a function (where the output value, y, is equal to zero) In our car example it is when the car no longer has any value. How realistic (to you) is this value? How realistic to your insurance company is this value?

Activity Continued Questions: Determine the n-intercept of the graph of the car value What is the practical significance of the n-intercept? Years of ownership Value in Dollars20,02518,64017,25515,87013,1008,945 y x 0 = – 1385x x = 20025/1385 = The car's worth nothing after 14 ½ years of ownership

Celsius and Fahrenheit Identify the slope, y-intercept and x-intercept of the line F = 1.8C + 32 and graph it. F x (C, F) (0, 32) F-intercept (17.78, 0) C-intecept slope = 1.8

Summary and Homework Summary –Slope Intercept form of Line: y = mx + b –Y-intercept (0,b) is The point where the graph crosses the y-axis Found by setting x = 0 in the slope intercept form –X-intercept (a,0) is The point where the graph crosses the x-axis Also known as the zero of the function Found by setting y = 0 in the slope intercept form –Increasing functions have a positive slope –Decreasing functions have a negative slope Homework –Pg ; 1 - 8