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To solve each part of the problem, we need to use the given quadratic equation \( y = a(x - r)(x - s) \) and apply the conditions provided in each subpart. Let's go through each one step-by-step:

### Part (a)

**Given:** The parabola has zeros at \((4, 0)\) and \((2, 0)\) and a y-intercept at \((0, 1)\).

**Equation form:** \( y = a(x - 4)(x - 2) \).

To find \( a \), we use the y-intercept \((0, 1)\):
\[ 1 = a(0 - 4)(0 - 2) \]
\[ 1 = a \cdot (-4) \cdot (-2) \]
\[ 1 = 8a \]
\[ a = \frac{1}{8} \]

### Part (b)

**Given:** The parabola has x-intercepts at \((4, 0)\) and \((-2, 0)\) and a y-intercept at \((0, -1)\).

**Equation form:** \( y = a(x - 4)(x + 2) \).

To find \( a \), we use the y-intercept \((0, -1)\):
\[ -1 = a(0 - 4)(0 + 2) \]
\[ -1 = a \cdot (-4) \cdot 2 \]
\[ -1 = -8a \]
\[ a = \frac{1}{8} \]

### Part (c)

**Given:** The parabola has zeros at \((5, 0)\) and \((0, 0)\) and a minimum value of \(-10\).

**Equation form:** \( y = a(x - 5)x \).

The vertex (minimum) of the parabola is at \( x = \frac{5 + 0}{2} = 2.5 \).

Using the minimum value \( -10 \) at \( x = 2.5 \):
\[ -10 = a(2.5 - 5)(2.5) \]
\[ -10 = a(-2.5)(2.5) \]
\[ -10 = a(-6.25) \]
\[ a = \frac{-10}{-6.25} = \frac{10}{6.25} = \frac{40}{25} = \frac{8}{5} \]

### Part (d)

**Given:** The parabola has x-intercepts at \((5, 0)\) and \((-3, 0)\) and a maximum value of 6.

**Equation form:** \( y = a(x - 5)(x + 3) \).

The vertex (maximum) of the parabola is at \( x = \frac{5 + (-3)}{2} = 1 \).

Using the maximum value \( 6 \) at \( x = 1 \):
\[ 6 = a(1 - 5)(1 + 3) \]
\[ 6 = a(-4)(4) \]
\[ 6 = a(-16) \]
\[ a = \frac{6}{-16} = -\frac{3}{8} \]

### Part (e)

**Given:** The parabola has its vertex at \((5, 0)\) and a y-intercept at \((0, -10)\).

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. Here, the vertex is at \((5, 0)\):
\[ y = a(x - 5)^2 \].

To find \( a \), we use the y-intercept \((0, -10)\):
\[ -10 = a(0 - 5)^2 \]
\[ -10 = a(25) \]
\[ a = \frac{-10}{25} = -\frac{2}{5} \]

These are the values of \( a \) for each condition:

- (a) \( a = \frac{1}{8} \)
- (b) \( a = \frac{1}{8} \)
- (c) \( a = \frac{8}{5} \)
- (d) \( a = -\frac{3}{8} \)
- (e) \( a = -\frac{2}{5} \)

Do you want details or have any questions?

Here are some related questions for further practice:
1. How do you find the vertex of a parabola given its standard form equation?
2. What are the steps to convert a quadratic equation from standard form to vertex form?
3. How do you determine the direction in which a parabola opens?
4. What is the significance of the coefficient \( a \) in a quadratic equation?
5. How can you find the axis of symmetry for a parabola given its equation in standard form?
6. What methods can be used to find the roots of a quadratic equation?
7. How does the value of \( a \) affect the width of the parabola?
8. What is the relationship between the discriminant and the nature of the roots of a quadratic equation?

**Tip:** When solving quadratic equations, always check the given conditions (like intercepts, vertex, etc.) to determine the correct form of the equation and values of the coefficients.

Explore how to solve quadratic equations under different conditions such as intercepts and vertex positions. Detailed solutions for each scenario provided with step-by-step explanations. Suitable for high school students learning algebra and quadratic equations.

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