Rate of Change & Slope Essential Question? How do you find the rate of change (slope)? 8.F.4

Common Core Standard: 8.F.4 ─ Use functions to model relationships between quantities. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Objectives: To find the slope (rate of change) of a linear function.

Curriculum Vocabulary Rate of Change (tasa de cambio): The ratio of vertical change (change in the dependent variable, y) to horizontal change (change in independent variable, x) in a function; the rate at which the quantity represented by y increases or decreases with respect to a change in the quantity represented by x. Initial Value (valor inicial): The starting value of a function; the first pair of x, y values for which a function is true.

Finding the Rates of Change Step 1: Identify the independent and dependent variables. (input) (output) Step 2: Find the 𝒐𝒖𝒕𝒑𝒖𝒕 𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏𝒑𝒖𝒕 𝒄𝒉𝒂𝒏𝒈𝒆 for each jump on the table. Step 3: Simplify the fraction

Finding the Rates of Change Find the rate of change for the following table: Jump 1: 30 2 =15 Jump 2: 45 3 =15 Jump 3: Input Output 1 15 3 45 6 90 8 120 30 2 45 3 30 2 Is the rate of change constant? YES

Finding the Rates of Change Find the rate of change for the following table: Jump 1: 18 0.5 =36 Jump 2: 13 1 =13 Jump 3: −5 0.5 =-10 Input Output 0.5 18 1.5 31 2 26 18 0.5 13 1 -5 0.5 Is the rate of change constant? NO When the rate of change is NOT constant, we say there is a VARIABLE rate of change.

𝒎= 𝒓𝒊𝒔𝒆 𝒓𝒖𝒏 = 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 = 𝚫𝒚 𝚫𝒙 SLOPE Slope(pendiente): The slope of a line is the ratio of the change in y-values (rise) to the corresponding change in x-values (run). Slope uses the letter m. 𝒎= 𝒓𝒊𝒔𝒆 𝒓𝒖𝒏 = 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 = 𝚫𝒚 𝚫𝒙 http://stevensonalgebra.weebly.com/algebra-songs.html

Undefined Slope (No Slope) KINDS of SLOPE There are four kinds of slope. Positive slope Negative slope Zero Slope Undefined Slope (No Slope)

Positive Slope Slope = Rise over Run Positive slope goes uphill These ratios are always positive.

Negative Slope Slope = Rise over Run Negative slope goes downhill These ratios are always negative.

Zero Slope Slope = Rise over Run HORIZONTAL LINE. These slope are always zero.

Undefined Slope (no slope) Slope = Rise over Run Undefined slope looks like a cliff (VERTICAL LINE) These ratios are always have 0 in the denominator which is UNDEFINED.

Slope Practice: Identify the type.

How to Find Slope 1) Look at a graph and count RISE over RUN.

Slope from a Graph Slope - The rate of change that determines the direction of a line and how steep it is. y x vertical change Slope = horizontal change +2 +3 -6 rise Slope = run -4 +3 3 2 = Slope = m = +2 -6 3 m = = -4 2

Find the slope of the line below. x y Slope = rise run +1 Slope = +4 +4 m = 1 4 +1 -2 -2 = 1 4 -8 m = -8

Find the slope of the line below. x y Slope = rise run +1 +3 Slope = +3 +1 m = 3

Find the slope of the line below. x y +4 Slope = rise run +3 +3 Slope = +4 m = 3 4

Find the slope of the line below. x y Slope = rise run -1 +2 -1 Slope = +2 m = 1 2

Find the slope of the line below. x y Slope = rise run -2 -2 Slope = +3 +3 m = 2 3

Find the slope of the line below. x y Slope = rise run -2 Slope = +1 m = -2 -2 +1

Identify from a graph Identify the type of slope. Find the slope.

Identify from a graph Identify the type of slope. Find the slope.

Identify from a graph Identify the type of slope. Find the slope.

Identify from a graph Identify the type of slope. Find the slope.

Identify from a graph Identify the type of slope. Find the slope.

Identify from a graph Identify the type of slope. Find the slope.

USING THE SLOPE FORMULA Step 1: Identify two points on the line. Point 1: (2,6) Point 2: (3,9) Step 2: LABEL each point (x1, y1) Point 1: (3,9) (x2, y2) Step 3: Write the slope formula 𝒎= 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 Step 4: Substitute and solve (SIMPLIFY!) 𝒎= 𝟗−𝟔 𝟑−𝟐 = 𝟑 𝟏 =𝟑

Try These

Try These Find the slope of the line between the two points: (6,5) and (3,2) (-7,5) and (2,-4) (9,-7) and (-3,-7) (4,8) and (3,-2) (11,15) and (11,-2)