MPA-140: Slope Remember, slope is all about a line and how steep it is, and lines grow or shrink in TWO dimensions, with x’s and y’s we can plot on a graph. Equations that have 2 variables, like 4x + 2y = 8, graph in a straight line. We will work with these types of linear equations in this lesson.

Standard Form Equation The equation from the previous slide, 4x + 2y = 8 is an equation in STANDARD form. Standard form is Ax + By = C , where A, B, C are all real numbers. If I pick INPUTS(x), then plug them into the equation 4x+2y=8, get some OUTPUTS(y), I can plot this on a coordinate plane ( graph). Let’s do this to see what it looks like.

Standard Form Equation Let’s graph 4x + 2y = 8. I get to choose INPUTS for x, I choose 0, 1, 2 Plug them into the equation one at a time and solve for y. (INPUT(x) in, gives OUTPUT(y)). 4(0) + 2y = 8 >>> SOLVE for y >>>> y = 4 4(1) + 2y = 8 >>> SOLVE for y >>>> y = 2 4(2) + 2y = 8 >>> SOLVE for y >>>> y = 0

Standard Form Equation Outputs Standard Form Equation Inputs 4(0) + 2y = 8 >>> SOLVE for y >>>> y = 4 4(1) + 2y = 8 >>> SOLVE for y >>>> y = 2 4(2) + 2y = 8 >>> SOLVE for y >>>> y = 0 We can put x,y’s into table, and then graph to get ALL the possible combinations of x’s and y’s that are solutions. y x x y ( x,y) 4 (0,4) 1 2 (1,2) (2,0)

Slope Intercept Form Graphing by choosing x inputs, and solving for y outputs always works, but there is an easier way. We can transform ANY standard equation like 4x + 2y = 8 into “slope intercept” form. Slope intercept form is y = mx+b where m is the slope and b is the y-intercept ( where the line crosses the y axis).

Slope Intercept Form, y=mx+b This IS Standard form Transforming 4x + 2y = 8 into “slope intercept” form looks like this. 4x + 2y = 8 2y = -4x + 8 2 2 y = -4x/2 + 8/2 y = -2x + 4 SO what’s so AWESOME about slope intercept form? Let’s LEARN! Now you KNOW why we spent so much time on equations!! -4x -4x You always need to reduce your fractions. -4/2 = -2 This IS Slope intercept form

Slope Intercept Form, y=mx+b We know our y=mx + b equation is y = -2x + 4 , how can it help us graph? Let’s break it down! y= mx + b y = -2x + 4 The m stands for the slope of the line, and you know from a previous lesson that slope = RISE over RUN In this example, our slope is -2. In RISE/RUN it looks like this. We always put whole #’s over 1. m = rise run m = -2 1 This RISE/RUN slope will help us graph our line in a second.

Slope Intercept Form, y=mx+b We know our y=mx + b equation is y = -2x + 4 , how can it help us graph? Let’s break it down! y= mx + b y = -2x + 4 Let’s LEARN! The b stands for the y intercept, where our line CROSSES or intersects with the y-axis. Having this info gives a starting point for our line. In this example, our y intercept is at 4 on the y=axis, or the point ( 0,4) Now we have a slope of -2/1 , and a y intercept point of ( 0,4), we can graph this line.

Slope Intercept Form, y=mx+b Equation: y = -2x + 4 , Slope is -2/1 , y-int.: 4 Remember, we graphed this line earlier and it looked like this. Let’s graph it using our slope intercept information and see what we get y x

Slope Intercept Form, y=mx+b Equation: y = -2x + 4 , Slope is -2/1 , y-int.: 4 We start by putting a point at 4 on the y-axis, our y intercept. THEN, we use our slope as a roadmap to the next point. -2/1 means DOWN 2, to the RIGHT 1 y x This is our next point, and we ONLY need two points to make a line. DRAW the line. y x It’s the same line, and you will find that working with slope intercept form is the easiest way to find slope, y-intercept, and to graph. Let’s compare to the 4x+2y=8 line we made earlier using INPUTS/OUTPUTS.

Let’s try one together. STOP!! Use your notes, use inverse operations and change this equation around before you LOOK, then graph it! Convert -3x - 6y = 12 into slope intercept form ( y=mx+b), and then use the slope and y-intercept to graph. STEP 1: Change -3x-6y = 12 into y=mx+b -3x - 6y = 12 - 6y = 3x + 12 -6 -6 y = 3x/-6 + 12/-6 y = -1/2x + (-2) OR y = -1/2x - 2 You always need to reduce your fractions. 3/-6 = -1/2 . ALSO, so we can use RISE/RUN, we don’t change slope to a decimal, it stays a fraction. 3x 3x

Let’s try one together. STEP 2: Identify slope and y-intercept from our equation. Slope intercept equation: y = -1/2x – 2 Y-intercept: -2 ( point (0,-2) on graph, our starting point) Slope : -1/2 ( DOWN 1, RIGHT 2 when graphing from y-intercept point) Now it’s EASY to graph. Put a point on (0, -2) and use our slope to find the next point.

Let’s try one together. x y STEP 3: Use slope and y-intercept to plot graph of possible x, y solutions. Slope intercept equation: y = -1/2x – 2 Y-intercept: -2 ( point (0,-2) on y-axis, our starting point.) Slope : -1/2 ( DOWN 1, RIGHT 2 when graphing, and the map to our 2nd point.) y x Once you have this line, you can figure out a lot about what’s happening with your data, or what will happen in the future by extending your line.

STOP!! Use your notes, use inverse operations and change this equation around before you LOOK, then graph it! Now YOU try one. Convert 4x - 6y = -12 into y=mx+b form, and then use the slope and y-intercept to graph. STEP 1: Change 4x-6y = -12 into y=mx+b 4x-6y = -12 -6y = -4x -12 -6 -6 y = -4x/-6 -12/-6 y = 2/3x + 2 -4x -4x You always need to reduce your fractions. -4/-6 = 2/3 . ALSO, so we can use RISE/RUN, we don’t change slope to a decimal, it stays a fraction.

YOU Try One STEP 2: Identify slope and y-intercept from our equation. Slope intercept equation: y = 2/3x + 2 Y-intercept: +2 ( point (0,2) on graph, our starting point.) Slope : 2/3 ( UP 2 , RIGHT 3 when graphing from y intercept) Now it’s EASY to graph. Put a point on (0, 2) and use our slope to find the next point.

You Try One. STEP 3: Use slope and y-intercept to plot graph of possible x, y solutions. Slope intercept equation: y = 2/3x+2 Y-intercept: +2 ( point (0,2) on graph) Slope : 2/3 ( UP 2 , RIGHT 3 when graphing from y intercept) y x Once you have this line, you can figure out a lot about what’s happening with your data, or what will happen in the future by extending your line.

Recognizing Slope Intercept Don’t make the MATH more difficult. Look to see if the equation is ALREADY in slope intercept form. Then there’s nothing to change. Let’s identify the slope and y-intercept for each equation. Examples: -2/3x + 7 = y y = 4x -6 y = -6 + 7x 4x + 3y = 12 Already in y=mx+b, just flip it around to y= -2/3x + 7 , Slope: -2/3 y-int: 7 Already in y=mx+b. Slope: 4 y-int: -6 Already in y=mx+b, just move terms to y= 7x - 6 , Slope: 7 y-int: -6 This is in standard form and needs inverses to move into y=mx+b form. 4x + 3y = 12 3y = -4x + 12 3y = -4/3x + 4 Slope: -4/3 y-int: 4

Recognizing Slope Intercept STOP!! Use your notes, decide if you’re in slope intercept form or need to do something to get there, and then ID Slope and y-int. Recognizing Slope Intercept Now you try these 4 to see if you can recognize equations already in slope intercept form, equations that are close, and equations that need complete transformations from standard form. Examples: 6 + y = 3x y = 8/3x - 14 y = 7 – 3/4x -3x + y = 24 Needs a small transformation by subtracting 6 to get y= 3x - 6 , Slope: 3 y-int: -6 Already in y=mx+b. Slope: 8/3 y-int: -14 Close to y=mx+b, flip around terms to get y= -3/4x + 7 , Slope: -3/4 y-int: 7 This is in standard form, and needs inverses to move into y=mx+b form. -3x + y = 24 y = 3x + 24 Slope: 3 y-int: 24

Writing a slope intercept equation from a graph Since y=mx+b is made up from the slope(m), and the y-intercept(b), if we figure these out from the graph, we can easily write the equation of the line. Let’s work backwards from the example we just did, and then we’ll try a couple new ones. Your job, FIND the equation of this line. y x

Writing a slope intercept equation from a graph To write this equation, we need slope(m), and the y-intercept(b). We can figure these out from the graph. y-intercept(b): slope(m): 2 First, the y-intercept is here at 2. So this is our b. y x

Writing a slope intercept equation from a graph To write this equation, we need slope(m), and the y-intercept(b). We can figure these out from the graph. y-intercept(b): slope(m): 2 2/3 To find slope, we pick any TWO points close to each other, and RISE/RUN to find slope. RUN of 3 y x m = rise run Rise of 2 m = 2 3

Writing a slope intercept equation from a graph To write this equation, we need slope(m), and the y-intercept(b). We now have these. y-intercept(b): 2 slope(m): 2/3 We have our m and our b, we can write this equation. y = 2/3x + 2. We already knew this one, but you can see it works, let’s try another one.

Writing a slope intercept equation from a graph STOP!! Use your notes, figure out the slope(m) and y intercept(b), and then write the equation before you go on. What is the equation of this line? To write this equation, we need slope(m), and the y-intercept(b). We can figure these out from the graph. y-intercept(b): slope(m): y x

Writing a slope intercept equation from a graph STOP!! Use your notes, figure out the slope(m) and y intercept(b), and then write the equation before you go on. Writing a slope intercept equation from a graph What is the equation of this line? To write this equation, we need slope(m), and the y-intercept(b). We can figure these out from the graph. y-intercept(b): slope(m): To find slope, we pick TWO points close to each other, and RISE/RUN to find slope. -1 -3/5 First, the y-intercept is here at -1. So this is our b. y x m = rise run Rise of -3 m = -3 5 RUN of 5

Writing a slope intercept equation from a graph To write this equation, we need slope(m), and the y-intercept(b). We now have these. y-intercept(b): -1 slope(m): -3/5 We have our m and our b, we can write this equation. y = -3/5x - 1