Question

41.) given the graph with the points (-2, 0) and (0, 4), find the rate of change.
rate of change: _________ (1)
42.) graph the given linear equations.
a.) $y = 3x - 2$
b.) $2y = -4x - 6$
c.) $x = 3$
d.) $y = -1$

Explanation:

41)

Step1: Recall rate of change formula

The rate of change (slope) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Identify the points

We have the points \((-2, 0)\) and \((0, 4)\). So, \(x_1=-2\), \(y_1 = 0\), \(x_2=0\), \(y_2 = 4\).

Step3: Calculate the slope

Substitute the values into the formula: \(m=\frac{4 - 0}{0 - (-2)}=\frac{4}{2}=2\).

Step1: Find the y - intercept

For the equation \(y=mx + b\) (slope - intercept form), when \(x = 0\), \(y=-2\). So the y - intercept is \((0,-2)\).

Step2: Find another point using the slope

The slope \(m = 3=\frac{3}{1}\). From the y - intercept \((0,-2)\), move 1 unit to the right (increase \(x\) by 1) and 3 units up (increase \(y\) by 3). So we get the point \((1,1)\).

Step3: Plot the points and draw the line

Plot the points \((0,-2)\) and \((1,1)\) on the coordinate plane and draw a straight line passing through them.

b) Graph of \(2y=-4x - 6\) (first rewrite in slope - intercept form)

Step1: Rewrite the equation

Divide both sides of \(2y=-4x - 6\) by 2: \(y=-2x-3\).

Step2: Find the y - intercept

When \(x = 0\), \(y=-3\). So the y - intercept is \((0,-3)\).

Step3: Find another point using the slope

The slope \(m=-2=\frac{-2}{1}\). From the y - intercept \((0,-3)\), move 1 unit to the right (increase \(x\) by 1) and 2 units down (decrease \(y\) by 2). So we get the point \((1,-5)\).

Step4: Plot the points and draw the line

Plot the points \((0,-3)\) and \((1,-5)\) on the coordinate plane and draw a straight line passing through them.

c) Graph of \(x = 3\)

Answer:

2

42)

a) Graph of \(y = 3x-2\)