x | g(x)
4 | 6
8 | 3
12 | 0
16 | -3
5 answers
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):
Plugging these values into the slope formula:
\[ m = \frac{3 - 6}{8 - 4} = \frac{-3}{4} \]
So, the slope of the linear function is
\[ m = -\frac{3}{4} \]
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):
\[ 6 = -\frac{3}{4}(4) + b \]
\[ -\frac{3}{4} \times 4 = -3 \]
\[ 6 = -3 + b \]
\[ b = 6 + 3 = 9 \]
Thus, the y-intercept of the linear function is:
\[ b = 9 \]
In summary, the y-intercept is (0, 9) in coordinate form.
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:
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:
Thus, the slope is \(2\) and the y-intercept is \(6\).
