Day 3 – Graphs of Linear Equations

The variable t represents the time in minutes after the airplane begins its descent. The starting conditions is 24,000 feet. The constant rate of descent is -800 feet per minute because the plane must descend 24,000 feet in 30 minutes. The function 𝑦= 24,000 – 800t, or 𝑦=−800+24,000, models the descent. Recall from Lesson 8.1 that the equation 𝑦=800𝑡+24,000 is a linear function because its graph is a line. In fact, any equation of the form 𝑦=𝑚𝑥+𝑏 is a linear function. When 𝑏 is 0, 𝑦=𝑚𝑥+𝑏 becomes 𝑦=𝑚𝑥, which is a line that passes through the origin.

Graph the point A(3,6) on a coordinate plane Graph the point A(3,6) on a coordinate plane. Start with the equation 𝑦=𝒎𝑥. Guess and check numbers for 𝒎 by graphing the equations until you find a line that intersects point A. If you use a graphics calculator, enter point A(3, 6), and let your first guess for m be 4. Enter 𝑦=4𝑥, and then graph the function. See if the lines passes through A(3, 6).

If you use graph paper, substitute values for x in the equation 𝑦=4𝑥 to locate two points on the line, such as (0, 0) and (2,8). Then draw the line that connects the points. See if the line passes through 𝐴 3, 6 . When m is 4, the line 𝑦=4𝑥 is too steep to go through point A. Repeat the process with different values for m until you find an equation for the line that intersects point A.

Fitting the Line to a Point Repeat the process above for points 𝐵 through 𝐹. B(2,8) C(6, 3) D(3, -6) E(2, -8) F(4, 7) What is the connection between the coordinates of the given target point and the slope, m?

Fitting the Line to a Point The slope m, is the y-coordinate divided by the x-coordinate.

Changing the Slope Draw the graph of each equation on the same coordinate plane. Compare the graphs you get by varying the slope, m. 𝑦=𝑥 𝑦=5𝑥 𝑦=−2𝑥 𝑦= 1 2 𝑥 𝑦=− 1 3 𝑥 1. How are the graphs of these line alike? 2. How are the graphs of these lines different? 3. Guess what the graph of 𝑦=3𝑥 looks like. 4. Check your guess by drawing the graph. 5. Make a conjecture about how m, the coefficient of x, affects the graph of 𝑦=𝑚𝑥.

Changing the Slope

Changing the Slope 1. All pass through (0, 0) 2. Their slopes are different 3. Passes through (0, 0) with slope 3. 4.(graph) 5. The larger the absolute value of m, the steeper the line. Lines with positive slopes slope upward from left to right, and line with negative slope downward from left to right.

Introducing a Constant Draw the graphs of the next set of equations on the same axes. 𝑦=𝑥+3 𝑦=2𝑥+3 𝑦=5𝑥+3 𝑦=−2𝑥+3 1. How are the graphs of these lines alike? 2. How are the graphs of these lines different? 3. Guess what the graph of 𝑦=4𝑥+3 looks like. 4. Check your guess by graphing. 5. Make a conjecture about how adding 3 affects the graph of 𝑦=𝑚𝑥.

Introducing a Constant

Introducing a Constant 1.All cross y-axis at (0, 3) 2.Their slopes are different 3. Passes through (0, 3) with slope 4. 4. (graph) 5. Adding 3 raises the graph so that the y-intercept moves from (0, 0) to (0, 3).

Introducing a Constant Draw the graphs of the next set of equations on the same axes. 𝑦=2𝑥 𝑦=2𝑥+3 𝑦=2𝑥+5 𝑦=2𝑥−3 6. How are the graphs of these lines alike? 7. How are the graphs of these lines different? 8. Guess what the graph of 𝑦=2𝑥−4 looks like. 9. Check your guess by graphing. 10. Make a conjecture about how the value of 𝑏 affects the graph of 𝑦=2𝑥+𝑏.

Introducing a Constant

Introducing a Constant 6. All have the same slope. 7. Their y-intercepts are different 8. Parallel to other four lines, but with y-intercept at (0, -4) 9. (graph) 10. The value of 𝑏 is the y- coordinates of the point where the line crosses the y-axis. It raises or lowers the line.

Introducing a Constant Describe the graph of each equation below without graphing. Check the accuracy of your descriptions by graphing each. 𝑦=−3 𝑦=5𝑥−6 𝑦=5+6𝑥 𝑦= 1 2 𝑥+3 How do the values of 𝑚 and 𝑏 affect the graph of 𝑦=𝑚𝑥+ 𝑏? Discuss the results of what you have discovered from the exploration. m affects the steepness of the graph; b affects where the graph will cross the y-axis.