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Name: Period #: _______ Date: ____________________ Notes……Unit 2.5 Slope Objective: To make a connection between direct variation and slope. Resources Direct variation – A proportional relationship that creates a continuous straight line that goes through the y origin and where is a x constant. The equation is y = kx. Slope – The measure of steepness of a line (rise over run) A direct variation equation is y = kx. The “k” in this equation is called the constant of proportionality or the constant of variation. It is found by doing y the calculation . We can also recall that the rate of change is a ratio that x compares the change in the output (y) to the change in the input (x). Which is y the same as . Another term, slope, is the measure of the steepness of a line x y (rise (y) over run (x)) which is also . So, these terms are used x interchangeably. But, not all functions (or equations) are direct variation functions (y = kx). Remember the famous slope-intercept equation of a line, y = mx + b? The “m” in the equation is the same as the “k” in the direct variation function. But the “b” is the y-intercept in this equation and since the direct variation function has to go through the origin (0, 0), then the y-intercept is 0. x-intercept – where the graph crosses the x-axis. y-intercept – where the graph crosses the y-axis. To find the rate of change in a linear function that has the form y = mx + b, we still need to divide y by x, but in a different way. We call it the “change of y y divided by the change of x” which looks like: . x Example 1: In a table, we can find rate of change like this: Table y-intercept Equation of the line x y 0 3 2 6 4 9 6 12 8 15 10 18 Resources Example 2: In a graph, we can find rate of change like this: Graph y-intercept Equation of the line Slope Formula – the change of y over the change of x. m y2 y1 x2 x1 Point-slope formula – y – y1 = m(x – x1) Example 3: When given two points, we can find the rate of change like this: Given (2, 4) and (3, 10) Slope-intercept form – First find the slope using the slope formula: y = mx + b m y2 y1 = x2 x1 Second, choose one of these options to find the equation of the line: Function Notation – is essentially replacing "y = " in your equations with "f(x) =". Point-slope formula: y – y1 = m(x – x1) OR slope-intercept form: y = mx + b Example: Change y =2x + 1 to f(x) = 2x + 1 Example 4: f(x) = 2x + 6 when x = 2, 20, 200, -2 f(2) = ______________ f(20) = _____________ Assignment: Unit 2.5 Classwork (All problems) f(200) = ______________ f(-2) = __________