Let's break down the problem step by step.
1. Evaluation of [tex]\( r(x) \)[/tex] for different values of [tex]\( x \)[/tex]:
The given function [tex]\( r(x) \)[/tex] is:
[tex]\[ r(x) = x^2 + 2x - 5 \][/tex]
Let's evaluate [tex]\( r(x) \)[/tex] for each [tex]\( \lambda \)[/tex] value given in the table:
For [tex]\( \lambda = 0 \)[/tex]:
[tex]\[ r(0) = 0^2 + 2 \cdot 0 - 5 = -5 \][/tex]
For [tex]\( \lambda = 1 \)[/tex]:
[tex]\[ r(1) = 1^2 + 2 \cdot 1 - 5 = 1 + 2 - 5 = -2 \][/tex]
For [tex]\( \lambda = 2 \)[/tex]:
[tex]\[ r(2) = 2^2 + 2 \cdot 2 - 5 = 4 + 4 - 5 = 3 \][/tex]
For [tex]\( \lambda = 3 \)[/tex]:
[tex]\[ r(3) = 3^2 + 2 \cdot 3 - 5 = 9 + 6 - 5 = 10 \][/tex]
For [tex]\( \lambda = 4 \)[/tex]:
[tex]\[ r(4) = 4^2 + 2 \cdot 4 - 5 = 16 + 8 - 5 = 19 \][/tex]
Thus, the evaluations of [tex]\( r(x) \)[/tex] for each value of [tex]\( \lambda \)[/tex] are:
- [tex]\( r(0) = -5 \)[/tex]
- [tex]\( r(1) = -2 \)[/tex]
- [tex]\( r(2) = 3 \)[/tex]
- [tex]\( r(3) = 10 \)[/tex]
- [tex]\( r(4) = 19 \)[/tex]
2. Summarizing [tex]\( \lambda \)[/tex], [tex]\( s(x) \)[/tex], and [tex]\( r(x) \)[/tex]:
We now have a set of values for [tex]\( \lambda \)[/tex], [tex]\( s(x) \)[/tex], and [tex]\( r(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\lambda & s(x) & r(x) \\
\hline
0 & -13 & -5 \\
1 & -3 & -2 \\
2 & 3 & 3 \\
3 & 5 & 10 \\
4 & 3 & 19 \\
\hline
\end{array}
\][/tex]
3. Intercepts of the function [tex]\( p \)[/tex]:
We know the function [tex]\( p \)[/tex] has the following intercepts:
- [tex]\( x \)[/tex]-intercept: at [tex]\((3, 0)\)[/tex]
- [tex]\( y \)[/tex]-intercept: at [tex]\((0, 6)\)[/tex]
Thus, the final numerical results and intercepts are summarized as follows:
### Combined Values Table
[tex]\[
\begin{array}{|c|c|c|}
\hline
\lambda & s(x) & r(x) \\
\hline
0 & -13 & -5 \\
1 & -3 & -2 \\
2 & 3 & 3 \\
3 & 5 & 10 \\
4 & 3 & 19 \\
\hline
\end{array}
\][/tex]
### Intercepts of Function [tex]\( p \)[/tex]
- [tex]\( x \)[/tex]-intercept: [tex]\((3, 0)\)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\((0, 6)\)[/tex]
These results give a complete picture of the evaluations and intercepts as required by the question.
