Let's go through the questions step by step:

1. Write a quadratic function given the following transformations:

• Reflection across the x-axis: This means multiplying the quadratic term by -1.
• Vertical shift right by 5 units: This implies shifting the function $x$ to $x-5$.
• Horizontal shift up by 3 units: This means adding 3 to the whole function.

The basic quadratic function is $y = x^2$. Applying the transformations step-by-step, we get:

1. Reflect across the x-axis: $y = -x^2$
2. Shift right by 5 units: $y = -(x - 5)^2$
3. Shift up by 3 units: $y = -(x - 5)^2 + 3$

So, the resulting quadratic function is: $y = -(x - 5)^2 + 3$

2. Write the equation of the linear function shown in the graph.

Looking at the graph:

• The red line appears to have two linear segments.
• To describe the equation of a line, we can use the slope-intercept form: $y = mx + b$.

1. For the segment on the left:
   • It appears to have a negative slope.
2. For the segment on the right:
   • It appears to descend as well.

However, further details from the graph would help specify exact slopes and intercepts. If you need a more detailed solution for this part, please clarify any additional data or points visible in the image.

Do you have any specific questions or want further elaboration on either part?

Related Questions:

1. How do transformations of the form $y = a(x - h)^2 + k$ affect the graph of a quadratic function?
2. What is the effect of a reflection across the y-axis compared to the x-axis for functions?
3. How can you determine the vertex of a quadratic function from its equation?
4. What role do slopes play in determining the equation of a linear function?
5. How can you identify if a function is linear or quadratic by looking at its graph?

Tip: For transformations involving shifts and reflections, always think of modifying each part of the basic function step-by-step to avoid mistakes.