This problem asks for a quadratic function, given three points:

• $x = 5$, $y = 15$
• $x = 0$, $y = 20$
• $x = 10$, $y = 0$

The general form of a quadratic function is:

$y = ax^2 + bx + c$

Let's solve step by step:

a) Determine the function

Substitute the points into the equation to form a system of equations.

1. For $x = 5$ and $y = 15$: $25a + 5b + c = 15$

2. For $x = 0$ and $y = 20$: $c = 20$

3. For $x = 10$ and $y = 0$: $100a + 10b + c = 0$

Since $c = 20$, substitute this into the other two equations:

1. $25a + 5b + 20 = 15 \Rightarrow 25a + 5b = -5$

2. $100a + 10b + 20 = 0 \Rightarrow 100a + 10b = -20$

Now, solve this system of equations:

1. $25a + 5b = -5$
2. $100a + 10b = -20$

Divide the second equation by 2:

$50a + 5b = -10$

Now subtract the first equation from this:

$(50a + 5b) - (25a + 5b) = -10 - (-5)$ $25a = -5$ $a = -\frac{1}{5}$

Now substitute $a = -\frac{1}{5}$ into $25a + 5b = -5$:

$25\left(-\frac{1}{5}\right) + 5b = -5$ $-5 + 5b = -5$ $5b = 0 \Rightarrow b = 0$

Thus, the quadratic function is:

$y = -\frac{1}{5}x^2 + 20$

b) What is the value of $x$ when $y$ is maximum?

For a quadratic function $y = ax^2 + bx + c$, the vertex (maximum or minimum point) is given by $x = -\frac{b}{2a}$.

Since $b = 0$, the vertex occurs at $x = 0$.

c) What is the value of $x$ when $x = y$?

Set $y = x$ in the quadratic equation:

$x = -\frac{1}{5}x^2 + 20$

Multiply everything by 5 to eliminate the fraction:

$5x = -x^2 + 100$ $x^2 + 5x - 100 = 0$

Solve this quadratic equation using the quadratic formula:

$x = \frac{-5 \pm \sqrt{5^2 - 4(1)(-100)}}{2(1)}$ $x = \frac{-5 \pm \sqrt{25 + 400}}{2}$ $x = \frac{-5 \pm \sqrt{425}}{2}$ $x = \frac{-5 \pm 20.62}{2}$

So, the solutions are:

$x_1 = \frac{-5 + 20.62}{2} = 7.81 \quad \text{and} \quad x_2 = \frac{-5 - 20.62}{2} = -12.81$

d) Graph the function

The quadratic function is $y = -\frac{1}{5}x^2 + 20$. It is an upside-down parabola with a maximum at $(0, 20)$ and intersects the x-axis at approximately $x = -10$ and $x = 10$.

Would you like a detailed plot of this function or further clarification?

Here are 5 additional related questions:

1. How does changing the coefficient $a$ in a quadratic function affect its graph?
2. How would you find the roots of a quadratic function using factoring instead of the quadratic formula?
3. What is the significance of the vertex in a quadratic function?
4. How do you calculate the discriminant of a quadratic equation, and what does it tell you about the solutions?
5. What happens to the quadratic function if the constant $c$ is increased or decreased?

Tip: Always check whether the parabola opens upwards or downwards by inspecting the sign of the coefficient $a$. A negative $a$ indicates it opens downwards (maximum point), and a positive $a$ means it opens upwards (minimum point).