Question

find the equation of the linear function represented by the table below in slope - intercept form.

x	y
2	-4
3	-7
4	-10

Explanation:

Step1: Find the slope (m)

We use the formula for slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Taking the points \((1, - 1)\) and \((2,-4)\), we have \( m=\frac{-4-(-1)}{2 - 1}=\frac{-4 + 1}{1}=-3 \). We can check with another pair, say \((2,-4)\) and \((3,-7)\): \( m=\frac{-7-(-4)}{3 - 2}=\frac{-7 + 4}{1}=-3 \). So the slope \( m=-3 \).

Step2: Find the y - intercept (b)

The slope - intercept form is \( y=mx + b \). We can use one of the points, say \((x = 1,y=-1)\) and \( m=-3 \). Substitute into the equation: \(-1=-3(1)+b\). Solve for \( b \): \(-1=-3 + b\), so \( b=-1 + 3 = 2\)? Wait, no, wait. Wait, when \( x = 1,y=-1 \), \( y=mx + b\) gives \(-1=-3\times1 + b\), so \( b=-1 + 3=2\)? But let's check with \( x = 2,y=-4 \): \( y=-3x + b\), \(-4=-3\times2 + b\), \(-4=-6 + b\), \( b=-4 + 6 = 2\). And with \( x = 3,y=-7 \): \(-7=-3\times3 + b\), \(-7=-9 + b\), \( b=-7 + 9 = 2\). Wait, but when \( x = 4,y=-10 \): \(-10=-3\times4 + b\), \(-10=-12 + b\), \( b=-10 + 12 = 2\). Oh, I made a mistake earlier in calculation. So \( b = 2\)? Wait, no, wait \( y=-3x + 2\). Let's check \( x = 1\): \( y=-3(1)+2=-1\), correct. \( x = 2\): \( y=-3(2)+2=-6 + 2=-4\), correct. \( x = 3\): \( y=-3(3)+2=-9 + 2=-7\), correct. \( x = 4\): \( y=-3(4)+2=-12 + 2=-10\), correct.

Answer:

The equation of the linear function in slope - intercept form is \( y=-3x + 2 \)