1. **State the problem:** We have a linear function graphed passing through points (0, -4) and (5, 10). We need to find its equation, the inverse function, sketch the inverse, create a table of values for the inverse, write the inverse equation, and find the intersection point of the function and its inverse. 2. **Find the equation of the original function:** - The slope $m$ is calculated by the formula $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - (-4)}{5 - 0} = \frac{14}{5} = 2.8.$$ - The y-intercept $b$ is the y-value when $x=0$, which is $-4$. - So, the equation is $$y = 2.8x - 4.$$ 3. **Find the inverse function:** - Start with $$y = 2.8x - 4.$$ - Swap $x$ and $y$: $$x = 2.8y - 4.$$ - Solve for $y$: $$x + 4 = 2.8y$$ $$y = \frac{x + 4}{2.8} = \frac{x + 4}{\frac{14}{5}} = \frac{5}{14}(x + 4).$$ - So, the inverse function is $$y = \frac{5}{14}x + \frac{20}{14} = \frac{5}{14}x + \frac{10}{7}.$$ 4. **Create a table of values for the inverse:** | x | y | |---|---| | -14 | $\frac{5}{14}(-14) + \frac{10}{7} = -5 + \frac{10}{7} = -5 + 1.4286 = -3.5714$ | | 0 | $\frac{5}{14}(0) + \frac{10}{7} = 0 + 1.4286 = 1.4286$ | | 14 | $\frac{5}{14}(14) + \frac{10}{7} = 5 + 1.4286 = 6.4286$ | 5. **Sketch the inverse:** The inverse is a line with slope $\frac{5}{14}$ and y-intercept $\frac{10}{7}$. It reflects the original line over the line $y=x$. 6. **Find the intersection point of the function and its inverse algebraically:** - Set original and inverse equal: $$2.8x - 4 = \frac{5}{14}x + \frac{10}{7}.$$ - Multiply both sides by 14 to clear denominators: $$14(2.8x - 4) = 14\left(\frac{5}{14}x + \frac{10}{7}\right)$$ $$39.2x - 56 = 5x + 20$$ - Rearrange: $$39.2x - 5x = 20 + 56$$ $$34.2x = 76$$ $$x = \frac{76}{34.2} = \frac{380}{171}$$ - Find $y$: $$y = 2.8x - 4 = 2.8 \times \frac{380}{171} - 4 = \frac{1064}{171} - 4 = \frac{1064}{171} - \frac{684}{171} = \frac{380}{171}.$$ 7. **Observation:** The intersection point is $$\left(\frac{380}{171}, \frac{380}{171}\right),$$ meaning the $x$ and $y$ coordinates are equal. This is expected because the function and its inverse intersect on the line $y = x$. **Final answers:** - Original function: $$y = 2.8x - 4$$ - Inverse function: $$y = \frac{5}{14}x + \frac{10}{7}$$ - Intersection point: $$\left(\frac{380}{171}, \frac{380}{171}\right)$$