Question

for #6 - 9, evaluate each of the following if x = 4 and y = 3.

1. $\frac{x - y}{x - y}$
2. $\frac{x + y}{y + x}$
3. $\frac{x - y}{y - x}$
4. $\frac{x - y}{y + x}$
5. if a, b, and c are all positive integers, which graph could represent the sketch of the graph of $p(x)=-a(x + b)(x^{2}-2cx + c^{2})$?
6. which of the following functions best models the graph pictured below?
7. $f(x)=(x + 1)^{2}(x - 2)$
8. $f(x)=(x - 1)^{2}(x + 2)$
9. $f(x)=(x + 1)(x - 2)^{2}$
10. $f(x)=(x - 1)(x + 2)^{2}$

Explanation:

Step1: Evaluate problem 6

Substitute \(x = 4\) and \(y=3\) into \(\frac{x + y}{x - y}\).
\[

$$\begin{align*} \frac{4+3}{4 - 3}&=\frac{7}{1}\\ &=7 \end{align*}$$

\]

Step2: Evaluate problem 7

Substitute \(x = 4\) and \(y = 3\) into \(\frac{x + y}{y + x}\). Since \(x + y=y + x\), \(\frac{4+3}{3 + 4}=\frac{7}{7}=1\).

Step3: Evaluate problem 8

Substitute \(x = 4\) and \(y = 3\) into \(\frac{x - y}{y - x}\). We have \(x - y=4 - 3 = 1\) and \(y - x=3 - 4=-1\), so \(\frac{4 - 3}{3 - 4}=\frac{1}{-1}=-1\).

Step4: Evaluate problem 9

Substitute \(x = 4\) and \(y = 3\) into \(\frac{x - y}{y + x}\). \(x - y=4 - 3 = 1\) and \(y + x=3 + 4 = 7\), so \(\frac{4 - 3}{3+4}=\frac{1}{7}\).

Step5: Solve problem 10

First, factor \(x^{2}-2cx + c^{2}=(x - c)^{2}\). Then \(p(x)=-a(x + b)(x - c)^{2}\). The roots of the polynomial are \(x=-b\) and \(x = c\) (with multiplicity 2). Since \(a>0\), the leading - coefficient of \(p(x)\) is negative. As \(x\to\pm\infty\), \(y\to-\infty\). The graph touches the \(x\) - axis at \(x = c\) (because of the multiplicity 2) and crosses the \(x\) - axis at \(x=-b\). The correct graph is (1).

Step6: Solve problem 11

The roots of the polynomial function can be found by setting \(y = 0\).
For \(f(x)=(x + 1)^{2}(x - 2)\), the roots are \(x=-1\) (with multiplicity 2) and \(x = 2\). When \(x=-1\), the graph touches the \(x\) - axis (due to multiplicity 2), and when \(x = 2\), the graph crosses the \(x\) - axis. The function \(f(x)=(x + 1)^{2}(x - 2)\) best models the given graph.

Answer:

1. 7
2. 1
3. -1
4. \(\frac{1}{7}\)
5. (1)
6. 1) \(f(x)=(x + 1)^{2}(x - 2)\)