1.5+Writing+Equations+for+Lines

Big Idea: Many real world situations can be modeled and predicted using mathematics. Essential Question: What kind of relationship makes straight lines on a graph? Notes:
 * August 26th, 2008**
 * M.P.**
 * Denise’s father pays her $5 each week.
 * Jonah’s mother paid him $20 at the beginning of the summer and now pays him $3 each week.

__Problem 1.5 Writing Equations for Lines:__

//In other words, which shows how Jonah’s earning relate to the number of weeks? Which graph model shows Denise’s earning? Explain how you know which graph is which.//
 * A.** //Which graph model shows Jonah’s earning as a function of the number of weeks?//

Jonah’s graph line is the orange one. This is because the orange line has a y-intercept of 20 and Jonah starts with $20, while the rate of change is 3 ($3). While the orange line is Jonah’s the purple line is Denise’s, because the purple line’s rate of change is 5 per week and Denise's father pays Denise $5 per week.


 * B.** //Write liner equations of the form y=Mx+B to show the relationships between Denise's and Jonah’s earnings and the number of weeks.//

Jonah’s Equation is Y= 3//x//+20 Denise’s Equation is Y= 5//x//


 * C.** //What do the values of **m** and **b** in each equation tell you about the relationship between the number of week s and dollars earned? What do the values of **m** and **b** tell you about each graph model?//

Equation M and B in the equations tell us two things.

M (The slope or the rate of increase) tell us how much money the person would get per week. B (Y-intercept or were the beginning point is) tells us how much money the person gets at the beginning of the job, in this case the amount of money the kids get from before cutting the grass.

Graph M and B in the graph tell us two things:

M, for the graph it is the rate of increase or also known as the slope. B, for the graph it is the y-intercept, were the line touches the Y axis.

__Problem 1.5 Follow Up:__

**1.** //How do you find an equation for the line with slope 1.5 that passes through the point (5, 9.5)? Because the slope is the value of M in Y=Mx+B, you know that the equation is the form y=1.5x+b. But how can find b?// **A.** //Substitute 5 for x and 9.5 for y in the equation y = 1.5x+b. Solve for b, and explain what the result tells you about the line. Use your result to help you write an equation for the line.// First you would find a point on the line, in this case it is 5, 9.5 and then insert it into the form of Y = Mx + B. Point: 5, 9.5 Equation: Y = 1.5x + B  Y = 1.5x + B   9.5 = 1.5x · 5 + B   9.5 = 7.5x + B   9.5 – 7.5 = 7.5x + B – 7.5 2 = B By solving the equation above we now know that B = 2. Or in other words that 2 = the Y-intercept.

Point: 2, 5 Equation: Y = 3x + B
 * B.** //Use your reasoning from part A to find the equation with the line of slope 3 that passes through the point (2, 5).//

Y = 3x + B 5 = 3x · 2 + B 5 = 6x + B 5 – 6 = 6x + B - 6 -1 = B


 * 2. A.** //Modify the reasoning you used in question 1 to find an equation for the line that passes through the points (2, 1) and (7, 11).//

Points: 2, 1 and 7, 11 Equation: ?

__11 – 1__ = 10/5 = 2 so the slope of the line is 2 7 – 2

Now we need to find B, the Y-intercept.

Y = 2x + B 11 = 2x · 7 + B 11 = 14x + B 11 – 14 = 14x + B – 14 -3 = B Now we know that -3 = the y-intercept.

So now we know those 2 things, we can now make the equation for this graph line. The equation would be Y = 2x + -3


 * B.** //use the reasoning you used in part a to find an equation for a line that passes through the points (1, 8) and (4, 2)//

Points: 1,8 and 4,2 Equation: ?

__2 – 8__ = -6/3 = -2 4 – 1

Y = -2x + B 8 = -2x · 1 + B 8 = -2x + B 8 + -2 = -2x + B + -2 10 = B

So now that we know those 2 things, we can now make the equation for the graph line. The equations is Y = -2x + 10


 * C.** //Compare the equations you wrote for parts A and B. How are they alike? How are they different?//

Well both parts A and B have the same form of equation, Y = Mx + B, so that tells us both A and B have straight linear graph lines.

//Give reasons for your choices.//
 * 3.** //Match each equation with its line in the diagram.//


 * A. //Y = 0.5x + 3 //**line II because it is a positive line
 * B. //Y = -0.5x + 6 //** line III because it is a negative line
 * C. //Y = 0.5x + 6//** line I because it is a positive line and has the same y-intercept as B
 * D. //Y = -0.5x + 3//** line IV because it has the same y-intercept as A

**4.** **A.** //Point A is on lines II and III. Its X-coordinate is. Calculate it’s Y-coordinate by substituting 3 for X in the equation for line II. Then, calculate its y-coordinate by substituting 3 for X in the equation for line III. What do you observe? Why does this happen// Line II  Y= 0.5x ·3 + 3 Y = 1.5x + 3 Y = 4.5 Line III Y = -0.5x ·3 + 6 Y = -1.5x + 6 Y = 4.5 I think both of the equations pass through the point 4.5, 3. This is because line III is going down while line II is going up antonyms the same past. Since 1.5 + 3 = 4.5 so does -1.5 + 6 = 4.5, so both points would hit antonyms 4.5, 3. **B.** //Abdul and Wendy did some calculations to find the Y-coordinate of point A. Which student did the calculations correctly? What error did the other student make? // ** Abdul’s Work ** Y = 0.5 X 3 + 3 Y = 0.5 X 6 Y = 3 ** Wendy’s Work ** Y = 0.5 X 3 + 3 Y = 1.5 + 3 Y = 4.5 Wendy’s work is correct. This is because Abdul did addition before multiplying, which he should have done.