For the linear function
f(x) = 5x + 3
(a) Determine the slope and y-intercept
(b) Use the slope and y-intercept to graph the linear function.
(c) Determine the average rate of change of the function
(d) Determine whether the linear function is increasing, decreasing, or constant.
a) by the "just look at it" theorem
slope = 5, y-intercept is 3
b) can't graph on here
c) average rate of change = slope = 5
d) the slope is positive, so the function is increasing
college algebra ?? mmmhh.
(a) To determine the slope and y-intercept of the linear function f(x) = 5x + 3:
The slope (m) is the coefficient of x, which in this case is 5.
Therefore, the slope is 5.
The y-intercept (b) is the value of y when x = 0.
Substituting x = 0 into the equation, we get:
f(0) = 5(0) + 3
f(0) = 0 + 3
f(0) = 3
Therefore, the y-intercept is 3.
(b) To graph the linear function using the slope and y-intercept:
- Plot the y-intercept, which is (0, 3), on the graph.
- Use the slope to find the next point on the line. Since the slope is 5 (which means for every increase of 1 in x, y increases by 5), move 1 unit to the right and 5 units up from the y-intercept. This gives us the point (1, 8).
- Continue this pattern to plot more points. For example, move 1 unit to the right and 5 units up from (1, 8), and plot the point (2, 13).
- Connect all the plotted points to create a straight line.
(c) The average rate of change of the function is equal to the slope.
In this case, the average rate of change is 5.
(d) The linear function f(x) = 5x + 3 is increasing because the slope is positive (5 > 0). If the slope were negative, the function would be decreasing, and if the slope were 0, the function would be constant.
To answer these questions about the linear function f(x) = 5x + 3, let's break it down step by step:
(a) Determining the slope and y-intercept:
In the linear function f(x) = mx + b, m represents the slope and b represents the y-intercept. For our given function f(x) = 5x + 3, the slope is 5 and the y-intercept is 3.
(b) Graphing the linear function using the slope and y-intercept:
To graph the linear function, you can plot points on a coordinate plane based on the slope and y-intercept. Start by plotting the y-intercept (0, b) on the y-axis, which in this case is (0, 3). Then, use the slope to find more points by using the fact that the slope is the "rise over run."
For example, for a slope of 5, you can go up 5 units and right 1 unit from the y-intercept to get the point (1, 8). Connect these points, and you will have a straight line that represents the linear function.
(c) Determining the average rate of change:
The average rate of change of a linear function can be found by calculating the difference in the y-values divided by the difference in the x-values between two points on the line. Since this is a linear function, the rate of change is constant.
Let's take two points on the line, such as (0, 3) and (1, 8). The difference in y-values is 8 - 3 = 5, and the difference in x-values is 1 - 0 = 1. So, the average rate of change of the function is 5/1 = 5.
(d) Determining whether the linear function is increasing, decreasing, or constant:
In this case, the slope is positive (5), which means the function is increasing. A positive slope indicates that as x increases, the corresponding y-values also increase. Therefore, the linear function f(x) = 5x + 3 is increasing.
By following these steps, you can determine the slope, y-intercept, graph the linear function, find the average rate of change, and determine whether it is increasing, decreasing, or constant.