I was clear on everything from the past lessons, except… FIRE UP!! FRIDAY Muddiest Point…. What concept(s) on over the last 2 classes and your last quiz were you unsure about? On the ½ sheet of paper, finish this sentence…. I was clear on everything from the past lessons, except…

Linear Function Review Sections 2-2, 2-3, 2-4 Pages 82-103

Objectives I can find slope by these methods: Between 2 points From a graph (rise over run) I can graph a linear line using slope and y-intercept I can manually graph points I can write an equation for a linear line

Slope – The BIG Picture The slope of a line is the steepness of the line. Slope can be positive, negative, zero, or undefined. We use the little letter (m) to represent slope in an equation y= mx + b

4 Types of Slope Positive Negative No Slope Undefined Slope See examples next slides

y-axis Positive Slope m > 0 x-axis Goes UP a Mountain 1 2 6 3 4 5 7 8 9 10 x-axis y-axis -2 -6 -3 -4 -5 -7 -8 -9 -1 Positive Slope m > 0 Goes UP a Mountain

y-axis Goes DOWN a Mountain x-axis Negative Slope m < 0 1 2 6 3 4 5 7 8 9 10 x-axis y-axis -2 -6 -3 -4 -5 -7 -8 -9 -1 Goes DOWN a Mountain Negative Slope m < 0

y-axis Horizontal Line x-axis Zero Slope m = 0 1 2 6 3 4 5 7 8 9 10 -2 -2 -6 -3 -4 -5 -7 -8 -9 -1 Horizontal Line Zero Slope m = 0

y-axis Vertical Line x-axis Undefined Slope m = #/0 1 2 6 3 4 5 7 8 9 10 x-axis y-axis -2 -6 -3 -4 -5 -7 -8 -9 -1 Vertical Line Undefined Slope m = #/0

Real World Slope Problems What REAL World things would not work correctly without slope?

Ramps for Various Purposes

The Roof on Buildings

Decking & Roads

Slope Slope can really be defined as the vertical change divided by the horizontal change. (Rise over Run) The slope of a line passing through two points (x1, y1), and (x2, y2) can be found using the following formula” m = =

GUIDED PRACTICE for Examples 1 and 2 Find the slope of the line passing through the given points. 3. (0, 3), (4, 8) SOLUTION Let (x1, y1) = (0, 3) and (x2, y2) = (4, 8). m = y2 – y1 x2 – x1 = 8 – 3 4 – 0 = 5 4 ANSWER 5 4

GUIDED PRACTICE for Examples 1 and 2 4. (– 5, 1), (5, – 4) SOLUTION Let (x1, y1) = (– 5, 1) and (x2, y2) = (5, – 4) m = y2 – y1 x2 – x1 = – 4 – 1 5 – (–5) = 1 2 – ANSWER 1 2 –

GUIDED PRACTICE for Examples 1 and 2 5. (– 3, – 2), (6, -2) SOLUTION Let (x1, y1) = (– 3, – 2) and (x2, y2) = (6, -2). m = y2 – y1 x2 – x1 = -2 –( – 2) 6 – (–3) = 9 ANSWER = 0

GUIDED PRACTICE for Examples 1 and 2 6. (7, 3), (7, -1) SOLUTION Let (x1, y1) = (7, 3) and (x2, y2) = (7, -1). m = y2 – y1 x2 – x1 = -1 – 3 7 – 7 = 4 – = Undefined ANSWER

Finding Slope on a Graph or Real Object You can also find the slope of a graphed line or real object by using rise/run Pick two points on the graph or object and then look at the rise and run between the points The units must be the same for rise and run

y-axis -7 x-axis 13 m = -7/13 1 2 6 3 4 5 7 8 9 10 -2 -6 -3 -4 -5 -7 -2 -6 -3 -4 -5 -7 -8 -9 -1 -7 13 m = -7/13

y-axis Run 10 Rise 9 x-axis m = 9/10 1 2 6 3 4 5 7 8 9 10 -2 -6 -3 -4 -2 -6 -3 -4 -5 -7 -8 -9 -1 Run 10 Rise 9 m = 9/10

Slope Intercept Format Recall from Algebra-1 y = mx + b m is the slope b is the y-intercept value

Looking at X and Y-Intercept (0,4) (3,0)

Finding x and y Intercepts The x-intercept is the x-coordinate of the point it crosses the x-axis. Likewise, the y-intercept is the y-coordinate of the point crossing the y-axis. The x-intercept is the value of x when y=0 The y-intercept is the value of y when x=0

Example 1 5x – 3y = 15 (Find x & y intercepts) x intercept is when y=0 x = 3 (So the x intercept is (3,0) y intercept is when x=0 5(0) – 3y = 15 -3y = 15 y = -5 (So the y intercept is (0,-5)

Graphing Example 1 (3,0) (0,-5)

Graphing with slope If we know the slope of a line (m) and at least one ordered pair on the line (x1, y1), then we can graph the line. First: Plot the known point Second: Use the slope (rise over run) to find more points Last: Connect the points with a straight line

Graph: m = -4, (3,4) 28

Graph line thru Point (-2,-5) with slope of 3/5 29

Graphing with Slope-Intercept Another method to graph quickly is to get the equation in Slope-Intercept Format This gives us the slope (m) and Y-Intercept (b)

Slope Intercept Form y = mx + b m is the slope of the line b is the y-intercept point

y-axis y=-3x + 2 x-axis 1 2 6 3 4 5 7 8 9 10 -2 -6 -3 -4 -5 -7 -8 -9 -2 -6 -3 -4 -5 -7 -8 -9 -1 y=-3x + 2

y-axis x-axis y=2/3x - 3 1 2 6 3 4 5 7 8 9 10 -2 -6 -3 -4 -5 -7 -8 -9 -2 -6 -3 -4 -5 -7 -8 -9 -1 y=2/3x - 3

Constant Linear Lines What do these look like???

Getting Slope-Intercept Format Many times the equation is not in Slope-Intercept Format y = mx + b The goal is to get y all by itself on the left side of the equation. Lets do some examples.

2x + 3y = 9 2x + 3y = 9 (Write equation) 3y = -2x + 9 (Move 2x to the right) y = -2/3x + 3 (Divide by 3)

3x = 4y + 12 3x = 4y + 12 (Write equation) 4y + 12 = 3x (Flip equation) 4y = 3x - 12 (Move 12 to the right) y = 3/4x – 3 (Divide by 4)

Parallel Lines Review Parallel Lines have the SAME SLOPE m1 = m2.

Perpendicular Lines Review Perpendicular Lines have NEGATIVE RECIPROCAL SLOPES

Finding New Slopes Given y = -3x + 4 What is the parallel slope? -3 What is the perpendicular slope? 1/3

Writing an Equation Need slope (m) Need y-intercept (b) Then just plug in: y = mx + b

Example Find the slope intercept form of the line with a slope of –2/3 and passes through point (-6,1) y = mx + b 1 = -2/3(-6) + b 1 = 4 + b b = -3 y = -2/3x -3

Homework WS 1-3