find an equation of the linear function f using the given information. please show all steps and work
f(-1)=1, f(1)= -2
f(x1)=y1
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f(-1)=1
So x1=-1 and y1=1
f(x2)=y2
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f(1)=-2
So x2=1 and y2=-2
We have two points: (x1, y1) and (x2, y2) = (-1, 1) and (1, -2).
The slope between those two points is (y2-y1)/(x2-x1) = (-2-1)/(1-(-1)) = -3/2 = m
The equation for a line is (y-y1) = m(x-x1). Plug in y1, m, and x1 to get the answer.
[When you solve for y, you should get y=-3/2x-1/2]
h t t p : / / b i t . l y / b z p B K 9
To find the equation of a linear function, we need two points on the graph of the function. In this case, we have two points: (-1, 1) and (1, -2).
The equation of a linear function is written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
Step 1:
First, we need to find the slope of the line using the given points. The slope (m) can be found using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points.
Let's use the points (-1, 1) and (1, -2) to find the slope:
m = (-2 - 1) / (1 - (-1))
m = (-2 - 1) / (1 + 1)
m = -3 / 2
So, the slope of the linear function is -3/2.
Step 2:
Now that we have the slope, we can use one of the given points and the slope to find the y-intercept (b).
Let's use the point (-1, 1):
y = mx + b
1 = (-3/2)(-1) + b
1 = 3/2 + b
To simplify, let's convert 3/2 to the common denominator:
1 = 3/2 + b
1 = 3/2 + (2b/2)
1 = (3 + 2b) / 2
To isolate b, we multiply both sides by 2:
2(1) = 3 + 2b
2 = 3 + 2b
Now, subtract 3 from both sides:
-1 = 2b
To solve for b, divide both sides by 2:
-1/2 = b
So, the y-intercept (b) of the linear function is -1/2.
Step 3:
Now that we know the slope (m = -3/2) and the y-intercept (b = -1/2), we can write the equation of the linear function (f):
y = mx + b
y = (-3/2)x + (-1/2)
y = -3x/2 - 1/2
Therefore, the equation of the linear function f is: f(x) = -3x/2 - 1/2.