https://file.aiquickdraw.com/mathbot/file/172579863679294b8u0vj.jpg

I can see the image you uploaded contains a set of mathematical problems, likely from a past exam or exercise sheet. Let's break down the visible portion:

The diagram represents the graph of a function \( f(x) = -(x - 1)(x - 5) \), with points of intersection on the \( x \)-axis at \( x = 1 \) and \( x = 5 \), and a turning point between these two points.

Here’s a list of the problems in the image:

1. **5.1.1**: Write down the equations of the asymptotes of \( f \).
2. **5.1.2**: Write down the coordinates of \( B \) (likely one of the points on the graph).
3. **5.1.3**: Write down the equation of the tangent to \( f \) at \( x = 3 \).
4. **5.1.4**: Calculate the length of line \( AB \).
5. **5.1.5**: Determine the values of \( x \) for which \( f \) is decreasing.
6. **5.2.1**: Calculate the average gradient between points \( A \) and \( B \).
7. **5.2.2**: Determine the equation by reflecting the function \( f(x) \) about the \( x \)-axis.
8. **5.3**: Given \( g(x) = f(x) + 2 \), write down the equation of the turning point of \( g \).

Would you like me to solve any of these problems or provide further details? Here's a list of expanded questions related to the image:

1. What are asymptotes in mathematical functions, and how are they found for quadratic functions like \( f(x) \)?
2. How do you calculate the coordinates of a point on a graph, such as point \( B \)?
3. What is the formula for the tangent line to a function at a given point?
4. How can we calculate the length between two points \( A \) and \( B \) on a graph?
5. What does it mean for a function to be decreasing on an interval?

**Tip**: When solving problems involving turning points, it is helpful to first take the derivative of the function and set it to zero to find critical points.

Given the quadratic function f(x) = -(x - 1)(x - 5), answer the following questions related to its graph and properties.

Solve problems related to the quadratic function f(x) = -(x - 1)(x - 5). Learn how to calculate asymptotes, the equation of a tangent line, the distance between two points, and how to determine intervals of decrease. Suitable for Grade 11 students.

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