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Find the slope of the linear function.

x | y
-2 | 16
0 | 21
2 | 26
4 | 31

3 answers

To find the slope of a linear function given points, you can use the formula for the slope \( m \):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

You can choose any two points from the table. Let's use the points (0, 21) and (2, 26):

  1. Assign the points as follows:

    • \( (x_1, y_1) = (0, 21) \)
    • \( (x_2, y_2) = (2, 26) \)

  2. Plug the values into the slope formula:

\[ m = \frac{26 - 21}{2 - 0} = \frac{5}{2} \]

Thus, the slope of the linear function is \( \frac{5}{2} \). You can verify by calculating the slope using other pairs of points, and you should arrive at the same result.

Now what is the y- intercept?

To find the y-intercept of the linear function, we can use the general equation of a linear function in slope-intercept form, which is:

\[ y = mx + b \]

Here, \(m\) is the slope and \(b\) is the y-intercept. We already calculated the slope \(m = \frac{5}{2}\).

Using one of the points from the table, we can substitute the \(x\) and \(y\) values to solve for \(b\). Let's use the point (0, 21):

  1. Substitute \(x = 0\) and \(y = 21\) into the equation:

\[ 21 = \frac{5}{2}(0) + b \]

  1. This simplifies to:

\[ 21 = 0 + b \]

Thus, \(b = 21\).

The y-intercept of the linear function is \(21\). In summary, the equation for the linear function is:

\[ y = \frac{5}{2}x + 21 \]