Take a look at the following table:
x -2 -1 0 1 2
f(x) -5 -2 1 4 7
a. Write out an equation for f(x). Assume the function is linear
b. What is the slope? Is it negative or positive? What information do we get from the slope?
To find the equation for the function f(x), we can use the formula for the equation of a straight line, which is:
y = mx + b
where:
- y is the value of the function f(x)
- x is the input value
- m is the slope of the line
- b is the y-intercept (the value of y when x is zero)
To determine the equation for f(x), we need to find the values of m and b.
Looking at the table you provided, we can see that when x = -2, f(x) = -5, and when x = 0, f(x) = 1. Using these two points, we can find the slope of the line:
slope (m) = (change in y) / (change in x)
= (f(x2) - f(x1)) / (x2 - x1)
where (x1, f(x1)) = (-2, -5) and (x2, f(x2)) = (0, 1)
So, the slope is:
m = (1 - (-5)) / (0 - (-2))
= (1 + 5) / (0 + 2)
= 6 / 2
= 3
Now that we know the slope is 3, we can substitute it into our equation:
y = 3x + b
To find the value of b, we can use any point from the table. Let's use x = 0, f(x) = 1 as before:
1 = 3(0) + b
1 = b
Therefore, the value of b is 1.
Now we have the equation for f(x) based on the given table:
f(x) = 3x + 1
So, the equation for f(x) is f(x) = 3x + 1.
As for the slope, we found it to be positive (3). The slope indicates how much the function f(x) changes with respect to x. In this case, it means that for every unit increase in x, the value of f(x) increases by 3 units.