Final answer:
To find the equation of a linear function given two points, calculate the slope and use one point to find the y-intercept. Once the slope and y-intercept are found, write the equation in slope-intercept form and then convert it to standard form. The equation for this function in standard form is 2x - 3y + 5 = 0.
Step-by-step explanation:
To find the equation of the linear function with the given values, we can start by recognizing that a linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept. We are given that f(4) = -1 and f(7) = -3. Using these two points, we can determine the slope m of the line:
m = (y2 - y1) / (x2 - x1) = (-3 - (-1)) / (7 - 4) = -2 / 3
Now that we have the slope, we can use one of the points to find the y-intercept b. Let's use the point (4, -1):
-1 = (4)(-2/3) + b
-1 = -8/3 + b
b = -1 + 8/3 = -3/3 + 8/3 = 5/3
The equation in slope-intercept form is f(x) = -2/3x + 5/3. To convert to standard form, we multiply by 3 to eliminate fractions:
3f(x) = -2x + 5
-2x + 3f(x) - 5 = 0
2x - 3f(x) + 5 = 0 (standard form requires positive x coefficient)
Substitute f(x) with y to maintain convention, we get 2x - 3y + 5 = 0 as the equation in standard form.