7. The function \( g(x)=-(x-5) 2+26 \) describes the height of a circus clown (in feet above the ground) \( x \) seconds after he is propelled into the air by his circus cannon. Use Desmos to find the intercepts, round to 3 decimal places (be sure you can also do this algebraically). The intercepts are: \( \qquad \) Explain in context what the intercept(s) of \( g(x) \) represent: 8. Use Desmos to find the point(s) of intersection of \( h(x) \) and \( g(x) \). The point(s) of intersection are: \( \qquad \) Explain in context what the point(s) of intersection represent. 9. Write an inequality that can be solved to find when the clown is above the flock of birds. Use Desmos to solve. Write the solution here: \( \qquad \) 10. Write an inequality that can be solved to find when is the flock of birds above the clown? Use Desmos to solve. Write the solution here: \( \qquad \)

Solve the quadratic equation by following steps: - step0: Solve using square roots: \(-\left(x-5\right)^{2}+26=0\) - step1: Add or subtract both sides: \(-\left(x-5\right)^{2}=-26\) - step2: Divide both sides: \(\frac{-\left(x-5\right)^{2}}{-1}=\frac{-26}{-1}\) - step3: Divide the numbers: \(\left(x-5\right)^{2}=26\) - step4: Simplify the expression: \(x-5=\pm \sqrt{26}\) - step5: Separate into possible cases: \(\begin{align}&x-5=\sqrt{26}\\&x-5=-\sqrt{26}\end{align}\) - step6: Move the constant to the right side: \(\begin{align}&x=\sqrt{26}+5\\&x=-\sqrt{26}+5\end{align}\) - step7: Rewrite: \(x_{1}=-\sqrt{26}+5,x_{2}=\sqrt{26}+5\) Let's solve the problems step by step. ### Problem 7: Finding the Intercepts of \( g(x) \) The function given is: \[ g(x) = -(x-5)^2 + 26 \] **Finding the x-intercepts:** To find the x-intercepts, we set \( g(x) = 0 \): \[ -(x-5)^2 + 26 = 0 \] Rearranging gives: \[ (x-5)^2 = 26 \] Taking the square root of both sides: \[ x - 5 = \pm \sqrt{26} \] Thus, we have: \[ x = 5 \pm \sqrt{26} \] Calculating the approximate values: \[ x_1 \approx 5 - \sqrt{26} \quad \text{and} \quad x_2 \approx 5 + \sqrt{26} \] Now, let's calculate these values to three decimal places: \[ x_1 \approx 5 - 5.099 \approx -0.099 \quad \text{and} \quad x_2 \approx 5 + 5.099 \approx 10.099 \] **Finding the y-intercept:** To find the y-intercept, we set \( x = 0 \): \[ g(0) = -(0-5)^2 + 26 = -25 + 26 = 1 \] **Intercepts:** - x-intercepts: \( x \approx -0.099, 10.099 \) - y-intercept: \( y = 1 \) ### Explanation of Intercepts - The x-intercepts represent the times (in seconds) when the clown is at ground level (height = 0 feet). - The y-intercept represents the height of the clown at \( t = 0 \) seconds, which is 1 foot above the ground. ### Problem 8: Finding the Points of Intersection of \( h(x) \) and \( g(x) \) Since the function \( h(x) \) is not provided, I cannot find the points of intersection without its equation. Please provide the equation for \( h(x) \) to proceed. ### Problem 9: Inequality for When the Clown is Above the Flock of Birds Assuming the height of the flock of birds is a constant \( b \), the inequality can be written as: \[ g(x) > b \] To solve this inequality, we need the value of \( b \). Please provide the height of the flock of birds. ### Problem 10: Inequality for When the Flock of Birds is Above the Clown Similarly, the inequality can be written as: \[ b > g(x) \] Again, we need the value of \( b \) to solve this inequality. Please provide the necessary information for \( h(x) \) and the height of the flock of birds to continue.