Question

period given the function below, identify the characteristics listed below. ( y=(x)(x - 1)(x+2) ) roots: ( y - )intercept: end behavior: as ( x\to-infty,y\to ) as ( x\to+infty,y\to ) when you have two ( x - )intercepts at the same location on the graph, that location is called a one ( x - )intercept at the graph goes through the point. however, when there is only one ( x - )intercept at the location, the graph only given the function below, identify the characteristics listed below. ( y=(x)(x - 3)(x+2) ) roots: ( y - )intercept: ( (0,0) ) end behavior: as ( x\to-infty,y\to ) as ( x\to+infty,y\to ) write an equation that represents the function shown below. the polynomial above is (circle one) even or odd the leading coefficient of the equation is (circle one) positive or negative write the equation below. ( y = ) over what interval is the graph above increasing? a. ( (-infty,-2) ) b. ( (-5,0) ) c. ( (-2,infty) ) d. ( (-0.5,0) ) a rocket being launched from a hill is modeled by the function ( y=-16x^{2}+44x + 80 ), it will explode at its maximum height before crashing into the graph. use the graph below to answer the questions below. what is the height of the hill? how long does it take the rocket to reach its maximum height? at what time does the rocket crash into the water? how high is the rocket when it explodes?

Explanation:

Step1: Analyze polynomial functions

For a polynomial function \(y = f(x)=(x)(x - 1)(x+2)\), to find the roots, we set \(y = 0\).
When \(x=0\), \(y = 0\); when \(x - 1=0\) (i.e., \(x = 1\)), \(y=0\); when \(x + 2=0\) (i.e., \(x=-2\)), \(y = 0\). So the roots are \(x=-2,0,1\).

Step2: Find y - intercept

To find the y - intercept, we set \(x = 0\) in the function \(y=(x)(x - 1)(x + 2)\). Substituting \(x = 0\) gives \(y=(0)(0 - 1)(0+2)=0\). So the y - intercept is \((0,0)\).

Step3: Determine end - behavior

The polynomial \(y=(x)(x - 1)(x + 2)=x^{3}+x^{2}-2x\) is a cubic (degree 3, odd - degree) polynomial with a positive leading coefficient (the coefficient of \(x^{3}\) is 1). As \(x\to+\infty\), \(y\to+\infty\) and as \(x\to-\infty\), \(y\to-\infty\).

Step4: Analyze the rocket - height problem

For a rocket - height function \(y=-16x^{2}+44x + 80\), which is a quadratic function of the form \(y = ax^{2}+bx + c\) with \(a=-16\), \(b = 44\), \(c = 80\). The x - coordinate of the vertex (maximum height) is given by \(x=-\frac{b}{2a}=-\frac{44}{2\times(-16)}=\frac{44}{32}=\frac{11}{8}=1.375\) seconds.
Substitute \(x = 1.375\) into \(y=-16x^{2}+44x + 80\):
\[

$$\begin{align*} y&=-16\times(1.375)^{2}+44\times1.375 + 80\\ &=-16\times1.890625+60.5+80\\ &=-30.25+60.5 + 80\\ &=110.25 \end{align*}$$

\]
To find when the rocket crashes into the water (\(y = 0\)), we solve the quadratic equation \(-16x^{2}+44x + 80=0\). Divide through by - 4: \(4x^{2}-11x - 20=0\). Using the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) with \(a = 4\), \(b=-11\), \(c=-20\), we have \(x=\frac{11\pm\sqrt{(-11)^{2}-4\times4\times(-20)}}{2\times4}=\frac{11\pm\sqrt{121 + 320}}{8}=\frac{11\pm\sqrt{441}}{8}=\frac{11\pm21}{8}\). We get \(x=\frac{11 + 21}{8}=4\) or \(x=\frac{11-21}{8}=-\frac{5}{4}\). Since time \(x\geq0\), the rocket crashes into the water at \(x = 4\) seconds.

Answer:

Roots of \(y=(x)(x - 1)(x + 2)\): \(x=-2,0,1\); Y - intercept: \((0,0)\); End - behavior: As \(x\to+\infty\), \(y\to+\infty\), as \(x\to-\infty\), \(y\to-\infty\); Rocket maximum height: \(110.25\) feet at \(x = 1.375\) seconds; Rocket crashes into water at \(x = 4\) seconds.