The distance between the highest point on the graph of `y =-(x-1) (x-3)` and the lowest point on the graph of `y =x(x-4)` is -

Options

A. `0`
B. `3`
C. `4`
D. `5`

Explanation

To find the distance between the highest point on the graph of \( y = -(x-1)(x-3) \) and the lowest point on the graph of \( y = x(x-4) \), we will follow these steps:

### Step 1: Find the highest point of the first graph \( y = -(x-1)(x-3) \)

1. Rewrite the equation:
\[ y = -(x^2 - 4x + 3) = -x^2 + 4x - 3 \]
This is a downward-opening parabola.

2. The vertex (highest point) of a parabola given by \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \):
- Here, \( a = -1 \) and \( b = 4 \).
- Calculate:
\[ x = -\frac{4}{2 \cdot -1} = 2 \]

3. Substitute \( x = 2 \) back into the equation to find the \( y \)-coordinate:
\[ y = -2^2 + 4(2) - 3 = -4 + 8 - 3 = 1 \]
So, the highest point is \( (2, 1) \).

### Step 2: Find the lowest point of the second graph \( y = x(x-4) \)

1. Rewrite the equation:
\[ y = x^2 - 4x \]
This is an upward-opening parabola.

2. The vertex (lowest point) can also be found using \( x = -\frac{b}{2a} \):
- Here, \( a = 1 \) and \( b = -4 \).
- Calculate:
\[ x = -\frac{-4}{2 \cdot 1} = 2 \]

3. Substitute \( x = 2 \) back into the equation to find the \( y \)-coordinate:
\[ y = 2^2 - 4(2) = 4 - 8 = -4 \]
So, the lowest point is \( (2, -4) \).

### Step 3: Calculate the distance between the two points

1. The highest point is \( (2, 1) \) and the lowest point is \( (2, -4) \).
2. The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \( x_1 = 2, y_1 = 1 \) and \( x_2 = 2, y_2 = -4 \):
\[ d = \sqrt{(2 - 2)^2 + (-4 - 1)^2} = \sqrt{0 + (-5)^2} = \sqrt{25} = 5 \]

### Final Answer
The distance between the highest point on the graph of \( y = -(x-1)(x-3) \) and the lowest point on the graph of \( y = x(x-4) \) is \( 5 \).

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