Write an equation for the linear function f with the given values.
f(1)=3, f(3)=4
you are given 2 points:
(1,3) and (3,4)
find the slope,
m = (4-3)/(3-1) = 1/2
then
y-3 = (1/2)(x - 3)
arrange it into one of the standard ways to write a straight-line
equation.
Sure, let's put the clown nose on and get to work!
To find the equation for the linear function f, we need to determine the slope (m) and the y-intercept (b). Once we have those values, we can use the equation y = mx + b.
Given that f(1) = 3 and f(3) = 4, we can use these two points to find the slope:
m = (y2 - y1) / (x2 - x1)
= (4 - 3) / (3 - 1)
= 1/2
Now, we need to find the y-intercept, which is the value of f(0). We can use the slope-intercept form of a line equation to solve for b:
f(x) = mx + b
3 = (1/2)(1) + b
3 = 1/2 + b
b = 3 - 1/2
b = 5/2
So, the equation for the linear function f is:
f(x) = (1/2)x + 5/2
Just remember, if you're having trouble with the equation, I can always juggle some numbers for you!
To write an equation for the linear function f, we need to find the slope and the y-intercept.
First, let's find the slope. The formula for slope is:
slope = (change in y) / (change in x)
We can use the values of f(1) and f(3) to find the change in y and change in x.
change in y = f(3) - f(1) = 4 - 3 = 1
change in x = 3 - 1 = 2
Therefore, the slope is 1/2.
Next, we need to find the y-intercept. We can use the formula:
y = mx + b
Since we know the slope is 1/2 and we have a point (1, 3), we can substitute these values to find the y-intercept:
3 = (1/2)(1) + b
3 = 1/2 + b
To isolate b, subtract 1/2 from both sides:
3 - 1/2 = b
6/2 - 1/2 = b
5/2 = b
So, the y-intercept is 5/2.
Putting it all together, the equation for the linear function f is:
f(x) = (1/2)x + 5/2
To find an equation for the linear function f, we can use the point-slope form of a linear equation.
The point-slope form is given by:
y - y₁ = m(x - x₁),
where (x₁, y₁) are the coordinates of a point on the line, and m is the slope of the line.
Given that f(1) = 3, we have the point (1, 3).
To find the value of m (the slope), we can use the second point given, f(3) = 4, which corresponds to the coordinates (3, 4).
Using these two points, we can substitute the values of (x₁, y₁) = (1, 3) and (x, y) = (3, 4) into the point-slope form equation:
y - 3 = m(x - 1).
Substitute the y value as 4 and the x value as 3:
4 - 3 = m(3 - 1).
1 = 2m.
Now, solve for m:
m = 1/2.
We have the slope m = 1/2, and we can use the coordinates (1, 3) to find the equation:
y - 3 = (1/2)(x - 1).
Simplify the equation:
y - 3 = 1/2x - 1/2.
Add 3 to both sides:
y = 1/2x + 5/2.
Therefore, the equation for the linear function f with the given values is f(x) = 1/2x + 5/2.