Chapter 3

Two functions, \(f\) and \(g,\) are related by the given equation. Use the numerical representation of \(f\) to make a numerical representation of \(\mathbf{g}\). \(g(x)=f(x)-10\) $$\begin{array}{cccccc}x & 0 & 5 & 10 & 15 & 20 \\\\\hline f(x) & -5 & 11 & 21 & 32 & 47\end{array}$$

Solve for the specified variable. $$ \boldsymbol{W}=\boldsymbol{I}^{2} \boldsymbol{R} \text { for } \boldsymbol{I} $$

Two functions, \(f\) and \(g,\) are related by the given equation. Use the numerical representation of \(f\) to make a numerical representation of \(\mathbf{g}\). \(g(x)=f(x-2)\) $$\begin{array}{rrrrrr}x & -4 & -2 & 0 & 2 & 4 \\\f(x) & 5 & 2 & -3 & -5 & -9\end{array}$$

Solve for the specified variable. $$ a^{2}+b^{2}=c^{2} \text { for } b $$

Two functions, \(f\) and \(g,\) are related by the given equation. Use the numerical representation of \(f\) to make a numerical representation of \(\mathbf{g}\). \(g(x)=f(x+50)\) $$\begin{array}{rrrrrr}x & -100 & -50 & 0 & 50 & 100 \\\f(x) & 25 & 80 & 120 & 150 & 100\end{array}$$

Solve for the specified variable. $$ S=4 \pi r^{2}+x^{2} \text { for } r $$

Two functions, \(f\) and \(g,\) are related by the given equation. Use the numerical representation of \(f\) to make a numerical representation of \(\mathbf{g}\). \(g(x)=f(x+1)-2\) $$\begin{array}{rrrrrrr}x & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline f(x) & 2 & 4 & 3 & 7 & 8 & 10\end{array}$$

Solve for the specified variable. $$ s=-16 t^{2}+100 t \text { for } t $$

Two functions, \(f\) and \(g,\) are related by the given equation. Use the numerical representation of \(f\) to make a numerical representation of \(\mathbf{g}\). \(g(x)=f(x-3)+5\) $$\begin{array}{rrrrrr}x & -3 & 0 & 3 & 6 & 9 \\\\\hline f(x) & 3 & 8 & 15 & 27 & 31\end{array}$$

Solve for the specified variable. $$ T^{2}-k T-k^{2}=0 \text { for } T $$

Two functions, \(f\) and \(g,\) are related by the given equation. Use the numerical representation of \(f\) to make a numerical representation of \(\mathbf{g}\). \(g(x)=f(-x)+1\) $$ \begin{array}{rrrrrr} x & -2 & -1 & 0 & 1 & 2 \\ f(x) & 11 & 8 & 5 & 2 & -1 \end{array} $$

Could a quadratic function have one real zero and one imaginary zero? Explain.

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 3 x^{2}=12 $$

$$ \begin{array}{rrrrrr} x & -4 & -2 & 0 & 2 & 4 \\ \hline f(x) & 5 & 8 & 10 & 8 & 5 \end{array} $$

Give an example of a quadratic function that has only real zeros and an example of one that has only imaginary zeros. How do their graphs compare? Explain how to determine from a graph whether a quadratic function has real zeros.

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 8 x^{2}-2=14 $$

The points \((-12,6),(0,8),\) and \((8,-4)\) lie on the graph of \(y=f(x)\). Determine three points that lie on the graph of \(y=g(x)\). \(g(x)=f(x)+2\)

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ x^{2}-2 x=-1 $$

The points \((-12,6),(0,8),\) and \((8,-4)\) lie on the graph of \(y=f(x)\). Determine three points that lie on the graph of \(y=g(x)\). \(g(x)=f(x)-3\)

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 6 x^{2}=4 x $$

The points \((-12,6),(0,8),\) and \((8,-4)\) lie on the graph of \(y=f(x)\). Determine three points that lie on the graph of \(y=g(x)\). \(g(x)=f(x-2)+1\)

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 4 x=x^{2} $$

The points \((-12,6),(0,8),\) and \((8,-4)\) lie on the graph of \(y=f(x)\). Determine three points that lie on the graph of \(y=g(x)\). \(g(x)=f(x+1)-1\)

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 16 x^{2}+9=24 x $$

The points \((-12,6),(0,8),\) and \((8,-4)\) lie on the graph of \(y=f(x)\). Determine three points that lie on the graph of \(y=g(x)\). \(g(x)=-\frac{1}{2} f(x)\)

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ x^{2}+1=x $$

The points \((-12,6),(0,8),\) and \((8,-4)\) lie on the graph of \(y=f(x)\). Determine three points that lie on the graph of \(y=g(x)\). \(g(x)=-2 f(x)\)

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 2 x^{2}+x=2 $$

The points \((-12,6),(0,8),\) and \((8,-4)\) lie on the graph of \(y=f(x)\). Determine three points that lie on the graph of \(y=g(x)\). \(g(x)=f(-2 x)\)

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 2 x^{2}+3 x=12-2 x $$

The points \((-12,6),(0,8),\) and \((8,-4)\) lie on the graph of \(y=f(x)\). Determine three points that lie on the graph of \(y=g(x)\). \(g(x)=f\left(-\frac{1}{2} x\right)\)

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 3 x^{2}+3=5 x $$

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 9 x(x-4)=-36 $$

Use transformations of graphs to model the table of data with the formula \(f(x)=a(x-h)^{2}+k .\) (Answers may vary.) Number of titles released for DVD rentals $$ \begin{array}{llllll} \text { Year } & 1998 & 1999 & 2000 & 2001 & 2002 \\ \text { Titles } & 2049 & 4787 & 8723 & 14,321 & 21,260 \end{array} $$

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ \frac{1}{4} x^{2}+3 x=x-4 $$

Use transformations of graphs to model the table of data with the formula \(f(x)=a(x-h)^{2}+k .\) (Answers may vary.) Average sales price of a home in thousands of dollars $$ \begin{array}{|cccccc} \hline \text { Year } & 1970 & 1980 & 1990 & 2000 & 2005 \\ \hline \text { Price } & 30 & 80 & 150 & 210 & 300 \end{array} $$

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ x\left(\frac{1}{2} x+1\right)=-\frac{13}{2} $$

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 4 x=6+x^{2} $$

U.S. Home Ownership The general trend in the percentage \(P\) of homes lived in by owners rather than renters between 1990 and 2006 is modeled by $$P(x)=0.00075 x^{2}+0.17 x+44$$ where \(x=0\) comesponds to \(1990, x=1\) to \(1991,\) and so on. Determine a function \(g\) that computes \(P\), where \(x\) is the actual year. For example, \(P(0)=44,\) so \(g(1990)=44\)

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 3 x^{2}=1-x $$

The function \(D\) defined by $$D(x)=2375 x^{2}+5134 x+5020$$ models AIDS deaths \(x\) years after 1984 . Write a formula \(g(x)\) that computes AIDS deaths during year \(x,\) where \(x\) is the actual year.

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ x(5 x-3)=1 $$

A baseball is dropped from a stadium seat that is 75 feet above the ground. Its height \(s\) in feet after \(t\) seconds is given by \(s(t)=75-16 t^{2} .\) Estimate to the nearest tenth of a second how long it takes for the baseball to strike the ground.

A baseball is thrown downward with an initial velocity of 30 feet per second from a stadium seat that is 80 feet above the ground. Estimate to the nearest tenth of a second how long it takes for the baseball to strike the ground.

Explain how to graph the reflection of \(y=f(x)\) across the \(x\) -axis. Give an example.

From 1984 to 1994 the equation $$D(x)=2375 x^{2}+5134 x+5020$$ modeled the cumulative number of AIDS deaths \(x\) years after \(1984 .\) Estimate the year when there were \(90,000\) deaths.

From 1984 to 1994 the cumulative number of AIDS cases can be modeled by the equation $$C(x)=3034 x^{2}+14,018 x+6400$$ where \(x\) represents years after \(1984 .\) Estimate the year when \(200,000\) AIDS cases had been diagnosed.

If the graph of \(y=f(x)\) undergoes a vertical stretch or shrink to become the graph of \(y=g(x),\) do these two graphs have the same \(x\) -intercepts? \(y\) -intercepts? Explain your answers.

The width of a rectangular computer screen is 2.5 inches more than its height. If the area of the screen is 93.5 square inches, determine its dimensions symbolically, graphically, and numerically. Do your answers agree?

If the graph of \(y=f(x)\) undergoes a vertical stretch or shrink to become the graph of \(y=g(x),\) do these two graphs have the same \(x\) -intercepts? \(y\) -intercepts? Explain your answers.