2.1 Write down an equation for the asymptote of \( g \). Calculate the coordinates of the turning point of f . 2.3 Sketch the graphs of \( f \) and \( g \) on the same set of axes, clearly showing ALL intercepts with the axes, turning points and asymptotes. 2.4 Write down the range of \( f \). 2.5 What transformation does the graph of \( y=f(x) \) undergo in order to obtain the graph of \( y=2 x^{2}-2 \) ?

To provide a comprehensive response, I need to clarify the functions \( f \) and \( g \) that you are referring to. However, I can guide you through the general steps to solve each part of your question based on common types of functions. ### 2.1 Asymptote of \( g \) To find the asymptote of a function \( g(x) \), you typically look for horizontal or vertical asymptotes. - **Horizontal Asymptote**: This occurs when \( x \) approaches infinity or negative infinity. For rational functions, you can find it by comparing the degrees of the numerator and denominator. - **Vertical Asymptote**: This occurs where the function is undefined, typically where the denominator equals zero. If you provide the specific function \( g(x) \), I can help you find the asymptote. ### 2.2 Turning Point of \( f \) To find the turning point of a function \( f(x) \): 1. **Find the derivative** \( f'(x) \). 2. **Set the derivative equal to zero** to find critical points: \( f'(x) = 0 \). 3. **Determine the second derivative** \( f''(x) \) to classify the critical points (concave up or down). If you provide the specific function \( f(x) \), I can help you calculate the turning point. ### 2.3 Sketching the Graphs To sketch the graphs of \( f \) and \( g \): 1. **Identify intercepts**: Set \( f(x) = 0 \) and \( g(x) = 0 \) to find x-intercepts. For y-intercepts, evaluate \( f(0) \) and \( g(0) \). 2. **Plot turning points**: Use the coordinates found in part 2.2. 3. **Draw asymptotes**: Use the equation found in part 2.1. 4. **Sketch the graphs**: Use the information gathered to sketch the curves, ensuring to show the behavior as \( x \) approaches the asymptotes. ### 2.4 Range of \( f \) The range of a function \( f(x) \) is the set of all possible output values. To find the range: 1. Analyze the function's behavior (e.g., maximum and minimum values). 2. Consider any restrictions on the output based on the function's form. If you provide the specific function \( f(x) \), I can help you determine its range. ### 2.5 Transformation to Obtain \( y = 2x^2 - 2 \) To transform \( y = f(x) \) into \( y = 2x^2 - 2 \): 1. **Vertical Stretch**: The factor of 2 indicates a vertical stretch by a factor of 2. 2. **Vertical Shift**: The \(-2\) indicates a downward shift by 2 units. If \( f(x) \) is a basic quadratic function like \( y = x^2 \), then the transformation can be described as: - Stretch vertically by a factor of 2. - Shift down by 2 units. If you provide the specific function \( f(x) \), I can give a more tailored transformation description. Please provide the specific functions \( f \) and \( g \) for more detailed assistance!