To find the equation of the linear function represented by the data in slope-intercept form (y = mx + b), we need to determine the slope (m) and the y-intercept (b).
First, let's analyze the data points in the table:
- (11, 99)
- (22, 1414)
- (33, 1919)
- (44, 2424)
Step 1: Calculate the Slope (m)
The slope (m) can be calculated using two points from the table. We can choose any two points; let's use (11, 99) and (44, 2424).
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2424 - 99}{44 - 11} \]
Calculating this:
\[ m = \frac{2325}{33} = 70.4545 \]
For easier calculation, let's round \( m \) to \( 70.45 \).
Step 2: Find the y-intercept (b)
Now we can use one of the points and the slope to find the y-intercept (b). Using the point (11, 99):
\[ y = mx + b \] \[ 99 = 70.45(11) + b \]
Calculating \( 70.45 \times 11 \):
\[ 70.45 \times 11 = 775.95 \]
Now, substituting that back into the equation:
\[ 99 = 775.95 + b \]
To solve for \( b \):
\[ b = 99 - 775.95 \] \[ b = -676.95 \]
Step 3: Write the Equation in Slope-Intercept Form
Now that we have the slope and y-intercept, we can write the equation:
\[ y = 70.45x - 676.95 \]
This is the equation of the linear function in slope-intercept form based on the data provided.
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