The image contains a set of mathematical problems related to graph sketching. Here’s how to approach each problem:

Problem 1: Sketch the graphs of the following functions

• $a. \ y = x^2 - 2x + 4$
• $b. \ y = x^3 + x^2 - 2x$
• $c. \ y = 2x^2 - 4x + 3$
• $d. \ y = -2x^2 - 4x + 3$

Problem 2: Sketch the graph of the function

• $a. \ y = \frac{(x - 2)(x - 3)}{x - 5}$
• $c. \ y = x^4 - 2$

Problem 3: Sketch the piecewise function

• ( f(x) = \begin{cases} x^2 + 1 & \text{if } x \leq -2 \ 2x - x^2 & \text{if } -2 < x \leq 1 \ x - 1 & \text{if } x > 1 \ \end{cases} )

Problem 4: Sketch the piecewise function

• ( f(x) = \begin{cases} x^2 - 1 & \text{if } x \leq -2 \ -2x^2 + 3 & \text{if } -2 < x \leq 1 \ x + 1 & \text{if } x > 1 \ \end{cases} )

Approach to Solve:

1. Quadratic Functions (Problem 1a and 1c):

   • These functions are parabolas. Determine their vertex and direction (upward or downward).

2. Cubic Functions (Problem 1b):

   • Cubic functions can have one or two turning points. Analyze the behavior as $x$ approaches positive and negative infinity.

3. Rational Function (Problem 2a):

   • This function may have vertical asymptotes where the denominator is zero (i.e., $x = 5$) and might intersect the x-axis where the numerator is zero.

4. Higher Degree Polynomial (Problem 2c):

   • The graph of $y = x^4 - 2$ is a quartic function. It is symmetric about the y-axis and has a minimum at $x = 0$.

5. Piecewise Functions (Problems 3 and 4):

   • Evaluate each piece of the function within the specified domain and match the segments to form the complete graph.

Would you like more details on solving any specific part of these problems, or would you like to explore the graphical solutions?

5 Related Questions:

1. How do you find the vertex of a quadratic function?
2. What is the process for determining asymptotes in rational functions?
3. How can you identify turning points in a cubic function?
4. What are the general shapes of quartic functions, and how do they differ from parabolas?
5. How do you ensure continuity when sketching piecewise functions?

Tip: Always start by identifying the key features of the function, such as intercepts, asymptotes, and vertex, before sketching the graph.