Questions: If s(x) is a linear function, where s(-3)=-4, and s(2)=5, determine the slope intercept equation for s(x), then find s(-2). The equation of the line is: s(-2)= If f(x) is a linear function, where f(-2)=-4, and f(4)=3, determine the slope intercept equation for f(x), then find f(5). The equation of the line is: f(-5)=

Solution Steps

To find the slope-intercept equation of a linear function given two points, we first calculate the slope using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Then, we use the point-slope form of the equation \( y - y_1 = m(x - x_1) \) to find the equation in slope-intercept form \( y = mx + b \). Finally, we substitute the given x-value into the equation to find the corresponding y-value.

To find the slope-intercept form of the linear function \( s(x) \), we first calculate the slope \( m_s \) using the points \( (-3, -4) \) and \( (2, 5) \):

\[ m_s = \frac{5 - (-4)}{2 - (-3)} = \frac{9}{5} = 1.8 \]

Next, we find the y-intercept \( b_s \) using the point-slope form:

\[ b_s = y_1 - m_s \cdot x_1 = -4 - 1.8 \cdot (-3) = -4 + 5.4 = 1.4 \]

Thus, the equation of the line is:

\[ s(x) = 1.8x + 1.4 \]

Now, we substitute \( x = -2 \) into the equation of \( s(x) \):

\[ s(-2) = 1.8 \cdot (-2) + 1.4 = -3.6 + 1.4 = -2.2 \]

Next, we find the slope \( m_f \) for the linear function \( f(x) \) using the points \( (-2, -4) \) and \( (4, 3) \):

\[ m_f = \frac{3 - (-4)}{4 - (-2)} = \frac{7}{6} \approx 1.1667 \]

We then calculate the y-intercept \( b_f \):

\[ b_f = y_1 - m_f \cdot x_1 = -4 - 1.1667 \cdot (-2) = -4 + 2.3334 \approx -1.6667 \]

Thus, the equation of the line is:

\[ f(x) = 1.1667x - 1.6667 \]

Finally, we substitute \( x = 5 \) into the equation of \( f(x) \):

\[ f(5) = 1.1667 \cdot 5 - 1.6667 \approx 5.8335 - 1.6667 \approx 4.1667 \]

Final Answer