Let's analyze the image.

The given rational function is:

$f(x) = \frac{x^2 - 3x + 2}{3x^2 - 8x + 4}$

This function has been factored as:

$f(x) = \frac{(x - 2)(x - 1)}{(3x - 2)(x - 2)}$

The conclusions drawn are:

1. The x-intercept of the function: This occurs when the numerator equals zero, i.e., $x = a$.

   • Solving $(x - 2)(x - 1) = 0$, we get $x = 2$ and $x = 1$, so the x-intercepts are $x = 1$ and $x = 2$.

2. Point of discontinuity: A point of discontinuity occurs when both the numerator and denominator share a common factor, which is then canceled out. Here, $x - 2$ is a common factor in both the numerator and denominator.

   • The point of discontinuity is at $x = 2$.

3. Vertical asymptote: This occurs when the denominator is zero, but the numerator is not. From the factored form, the vertical asymptote is where $3x - 2 = 0$, i.e., $x = \frac{2}{3}$.

4. Horizontal asymptote: The horizontal asymptote is determined by comparing the degrees of the numerator and denominator. Since the degrees are the same (both are quadratic), the horizontal asymptote is given by the ratio of the leading coefficients of the numerator and denominator. This gives:

$y = \frac{1}{3}$

Now, for Numerical Response #5, we need to find the values of $a$, $b$, $c$, and $d$.

• The x-intercept $a = 1$ (the conclusion uses the smallest x-intercept),
• The point of discontinuity $b = 2$,
• The vertical asymptote $c = \frac{2}{3}$,
• The horizontal asymptote $d = 3$ (since $y = \frac{1}{d}$).

Thus, the values of $a$, $b$, $c$, and $d$ are:

$a = 1, \quad b = 2, \quad c = \frac{2}{3}, \quad d = 3$

Would you like any further clarification or more detailed steps?

Related Questions:

1. How do you determine the x-intercepts from a rational function?
2. What causes a point of discontinuity in a rational function?
3. How do you find the vertical asymptote of a rational function?
4. What are the rules for finding horizontal asymptotes?
5. How does factoring help in analyzing rational functions?

Tip:

Always check for common factors in the numerator and denominator of a rational function to identify points of discontinuity and simplify the function for further analysis.