To determine the linear function in slope-intercept form (y = mx + b) based on the points provided, we can first identify the slope (m) using any two points provided on the line and then calculate the y-intercept (b).
Step 1: Calculate the slope (m)
The slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let's use the points \((-2, -3)\) and \((2, 5)\):
- \((x_1, y_1) = (-2, -3)\)
- \((x_2, y_2) = (2, 5)\)
Substituting into the slope formula:
\[ m = \frac{5 - (-3)}{2 - (-2)} = \frac{5 + 3}{2 + 2} = \frac{8}{4} = 2 \]
Step 2: Calculate the y-intercept (b)
We can use the slope \(m = 2\) and one of the points to find the y-intercept \(b\). We'll use the point \((0, 1)\):
The slope-intercept form is:
\[ y = mx + b \]
Substituting the point \((0, 1)\) into the equation:
\[ 1 = 2(0) + b \implies b = 1 \]
Final Equation
Now that we have both \(m\) and \(b\), we can write the equation of the line:
\[ y = 2x + 1 \]
So, the linear function in slope-intercept form is:
\[ y = 2x + 1 \]