Question

Find the slope of the linear function.

x | g(x)
4 | 6
8 | 3
12 | 0
16 | -3

Answer 1

What is the y - intercept?

Answer 2

Determine the slope and y-intercept of the line.

with m = 2 going through the point (-5, -4)

Answer 3

To find the slope of the linear function represented by the points given in the table, we can use any two of the points. The slope \( m \) is calculated using the formula:

\[
m = \frac{g(x_2) - g(x_1)}{x_2 - x_1}
\]

Let's use the points (4, 6) and (8, 3):

- \( x_1 = 4 \), \( g(x_1) = 6 \)
- \( x_2 = 8 \), \( g(x_2) = 3 \)

Plugging these values into the slope formula:

\[
m = \frac{3 - 6}{8 - 4} = \frac{-3}{4}
\]

So, the slope of the linear function is

\[
m = -\frac{3}{4}
\]

Answer 4

To find the y-intercept of the linear function, we need to use the slope we calculated and one of the points from the table. The general equation of a linear function can be written in slope-intercept form as:

\[
g(x) = mx + b
\]

where \( m \) is the slope and \( b \) is the y-intercept.

We already found the slope \( m = -\frac{3}{4} \). Now, we can use one of the points to find \( b \). Let's use the point (4, 6):

1. Substitute \( m \), \( x \), and \( g(x) \) into the equation:

\[
6 = -\frac{3}{4}(4) + b
\]

2. Calculate \( -\frac{3}{4}(4) \):

\[
-\frac{3}{4} \times 4 = -3
\]

3. Now substitute that value back into the equation:

\[
6 = -3 + b
\]

4. Solve for \( b \):

\[
b = 6 + 3 = 9
\]

Thus, the y-intercept of the linear function is:

\[
b = 9
\]

In summary, the y-intercept is (0, 9) in coordinate form.

Answer 5

To find the equation of a line with a given slope and a point it passes through, we can use the point-slope form of a linear equation, which is given by:

\[
y - y_1 = m(x - x_1)
\]

where \((x_1, y_1)\) is the point on the line and \(m\) is the slope.

Given:
- Slope \(m = 2\)
- Point \((-5, -4)\)

Substituting these values into the point-slope form:

\[
y - (-4) = 2(x - (-5))
\]

This simplifies to:

\[
y + 4 = 2(x + 5)
\]

Now, distribute the 2 on the right side:

\[
y + 4 = 2x + 10
\]

Next, isolate \(y\) by subtracting 4 from both sides:

\[
y = 2x + 10 - 4
\]
\[
y = 2x + 6
\]

Now we have the equation of the line in slope-intercept form \(y = mx + b\), where:

- The slope \(m = 2\)
- The y-intercept \(b = 6\)

Thus, the slope is \(2\) and the y-intercept is \(6\).