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

Understand the Problem

The question is asking to find the equation of a linear function based on the provided values in the table. We will analyze the x and y values to determine the slope and y-intercept to write the equation in slope-intercept form.

Answer

The equation of the linear function is $y = 9x + 5$.

Steps to Solve

Identify the points from the table

The table provides pairs of $(x, y)$ values:

(0, 5)
(1, 14)
(2, 23)
(3, 32)
(4, 41)

Calculate the slope (m)

The slope $m$ can be calculated using the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Using the points (0, 5) and (1, 14): $$ m = \frac{14 - 5}{1 - 0} = \frac{9}{1} = 9 $$

Find the y-intercept (b)

The y-intercept is the value of $y$ when $x = 0$. From our table, when $x = 0$, $y = 5$, so $b = 5$.

Write the equation in slope-intercept form

The slope-intercept form is given by: $$ y = mx + b $$ Substituting the values of $m$ and $b$: $$ y = 9x + 5 $$

The equation of the linear function is $y = 9x + 5$.

More Information

This equation represents a linear function where the slope is 9, indicating that for every increase of 1 in $x$, $y$ increases by 9. The y-intercept of 5 means the line crosses the y-axis at the point (0, 5).

Tips

Incorrect slope calculation: Ensure to use the correct points to find the slope. Using points too close together may lead to inaccurate results.
Forgetting the y-intercept: When writing the equation, make sure to include both the slope and y-intercept.

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