Question

1. based on the degree and the end - behavior of h(x) what must be true about the sign of the leading coefficient of the equation? justify your answer.
2. use the given data to find to write two equations you could use to find the values for the constants a and b in the expression for h(x).
3. find the value of a and c. show your work.
4. write the following statement in limit notation: “as x increases without bound, q(x) decreases without bound.”
5. write the following limit in verbal form: lim(x→ - ∞) p(x)=∞

Explanation:

18.

Step1: Recall end - behavior rules

For a polynomial \(h(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_0\), if \(n\) is even, when \(x\to\pm\infty\), \(y\to+\infty\) if \(a_n>0\) and \(y\to-\infty\) if \(a_n < 0\). If \(n\) is odd, when \(x\to+\infty\), \(y\to+\infty\) if \(a_n>0\) and \(y\to-\infty\) if \(a_n < 0\); when \(x\to-\infty\), \(y\to-\infty\) if \(a_n>0\) and \(y\to+\infty\) if \(a_n < 0\). Analyze the given end - behavior of \(h(x)\) to determine the sign of \(a_n\). For example, if as \(x\to+\infty\) and \(x\to-\infty\), \(h(x)\to+\infty\), then the degree \(n\) is even and the leading coefficient is positive. If as \(x\to+\infty\), \(h(x)\to+\infty\) and as \(x\to-\infty\), \(h(x)\to-\infty\), then the degree \(n\) is odd and the leading coefficient is positive. Justify based on these rules.

Step1: Substitute first point

Substitute \((x_1,y_1)\) into \(h(x)=ax + b\) to get \(y_1=ax_1 + b\).

Step2: Substitute second point

Substitute \((x_2,y_2)\) into \(h(x)=ax + b\) to get \(y_2=ax_2 + b\).

Step1: Substitute first point

Substitute \((x_1,y_1)\) into \(y = ax^2+bx + c\) to get \(y_1=ax_1^2+bx_1 + c\).

Step2: Substitute second point

Substitute \((x_2,y_2)\) into \(y = ax^2+bx + c\) to get \(y_2=ax_2^2+bx_2 + c\).

Step3: Substitute third point

Substitute \((x_3,y_3)\) into \(y = ax^2+bx + c\) to get \(y_3=ax_3^2+bx_3 + c\). Then solve the system of equations for \(a\), \(b\), and \(c\).

Answer:

The sign of the leading coefficient depends on the degree \(n\) of the polynomial \(h(x)\) and its end - behavior. If the degree \(n\) is even and \(h(x)\to+\infty\) as \(x\to\pm\infty\), the leading coefficient is positive. If the degree \(n\) is even and \(h(x)\to-\infty\) as \(x\to\pm\infty\), the leading coefficient is negative. If the degree \(n\) is odd and \(h(x)\to+\infty\) as \(x\to+\infty\) and \(h(x)\to-\infty\) as \(x\to-\infty\), the leading coefficient is positive. If the degree \(n\) is odd and \(h(x)\to-\infty\) as \(x\to+\infty\) and \(h(x)\to+\infty\) as \(x\to-\infty\), the leading coefficient is negative.

19.
Since no data is given, assume \(h(x)=ax + b\) and we have two data points \((x_1,y_1)\) and \((x_2,y_2)\).