Answer:
3) rate of change = -5/4
4) rate of change = 3
5) rate of change = 1
6) steepest slope = #4
7) Please see the attached graph and explanation below
8) Please see the attached graph and explanation below
Step-by-step explanation:
Note:
I will do questions 3, 4, 6, 7, and 8, and will let you work on question 5 (since it involves the same process of solving for the rate of change as questions 3 and 4). However, the rate of change for question 5 is 1.
3) Calculate the Rate of Change from the Graph:
The rate of change is essentially the same as the slope of a linear equation, where it represents the ratio of a change in y to a corresponding change in x.
In order to solve for the rate of change, choose two points from the given graph.
Let (x₁, y₁) = (0, 200)
(x₂, y₂) = (80, 100)
Substitute these values into the following slope formula:
m = (y₂ - y₁)/(x₂ - x₁)
m = (100 - 200)/(80 - 0)
m = -100/80
Reduce to lowest terms by dividing the numerator and the denominator by 20/20:
m = -5/4
Therefore, the rate of change is -5/4.
4) Calculate the Rate of Change from the Table:
Similar to what we did in question 3, choose two ordered pairs from the given table:
Let (x₁, y₁) = (0, -1)
(x₂, y₂) = (3, 8)
Substitute these values into the following slope formula:
m = (y₂ - y₁)/(x₂ - x₁)
[tex]m = \frac{8 - (-1)}{3 - 0}[/tex]
[tex]m = \frac{8 + 1 }{3}[/tex]
[tex]m = \frac{9}{3}[/tex]
m = 3
Therefore, the rate of change from the given table is 3.
6) Which linear function has the steepest slope?
The positive-sloped lines whose rate of change has the highest value will have the steepest slope. For lines with negative slopes (downward-tilting lines), the highest absolute value for its rate of change will have the steepest negative slope.
Hence, the linear function with the steepest slope is #4 because the value of y changes by 3 units for every unit of change in x values.
7) Graph the equation: y = -6x - 4:
The linear equation, y = -6x - 4, where the slope, m = -6, and the y-intercept is (0, -4).
- To graph this equation, start by plotting the y-intercept.
- Then, use the slope, m = -6/1 (down 6 units, run 1 unit to the right) to plot other points on the graph. In doing so, your next graph should occur at point, (1, -10). Connect the two points to create a line that will represent the given equation, y = -6x - 4.
Technically, two points are sufficient enough to connect and create a line with. Please see the attached screenshot of the graph for y = -6x - 4.
8) Graph the following equation: 3x - 4y = 12
The given linear equation is in its standard form, Ax + By = C. It is easier to graph when the equation is in its slope-intercept form, y = mx + b.
In order to transform the given standard equation to slope-intercept form, start by subtracting 3x from both sides:
3x - 4y = 12
3x -3x - 4y = -3x + 12
-4y = -3x + 12
Divide both sides by -4 to isolate y:
[tex]\frac{-4y}{-4} = \frac{-3x + 12}{-4}[/tex]
y = ¾x - 3 ⇒ This is the slope-intercept form where the slope, m = ¾, and the y-intercept is (0, -3).
Graph:
Similar to how we graphed the equation from question 7, start by plotting the y-intercept (0, -3) on the graph. Then, use the slope, m = ¾ (3 units up, 4 units run), to plot other points on the graph. Your next point occurs at the x-intercept, (4, 0). The x-intercept is the point on the graph where it crosses the x-axis. Connect the two points to create a line that will represent the given equation, 3x - 4y = 12.
Please see the attached screenshot of the graph for 3x - 4y = 12.
