Chapter 2

Find the vertex and axis of symmetry of the associated parabola for each quadratic function.Then find at least two additional points on the parabola and sketch the parabola by hand. $$f(x)=-2 x^{2}+4 x-1$$

Solve the radical equation to find all real solutions. Check your solutions. $$\sqrt{x+1}+2=x$$

In Exercises \(41-48,\) use \(f\) and \(g\) given by the following tables of values. $$\begin{array}{ccccc}x & -1 & 0 & 3 & 6 \\\\\hline f(x) & -2 & 3 & 4 & 2\end{array}$$ $$\begin{array}{ccccc}x & -2 & 1 & 2 & 4 \\\\\hline g(x) & 0 & 6 & -2 & 3\end{array}$$ Evaluate \(f(-1)\)

Find \(x+y, x-y, x y,\) and \(x / y\). $$x=4-5 i ; y=3+2 i$$

Use the verbal description to find an algebraic expression for the function. The graph of the function \(g(t)\) is formed by vertically scaling the graph of \(f(t)=t^{2}\) by a factor of -3 and moving it to the right by 1 unit.

Solve the quadratic equation by using the quadratic formula. Find only real solutions. $$x^{2}+2 x-1=0$$

Decide if each function is odd, even, or neither by using the definitions. $$f(x)=x^{5}-2 x$$

Find the vertex and axis of symmetry of the associated parabola for each quadratic function.Then find at least two additional points on the parabola and sketch the parabola by hand. $$f(x)=x^{2}-6 x+1$$

The height of a ball that is thrown directly upward from a point 200 feet above the ground with an initial velocity of 40 feet per second is given by \(h(t)=-16 t^{2}+40 t+200,\) where \(t\) is the amount of time elapsed since the ball was thrown; \(t\) is in seconds and \(h(t)\) is in feet. For what values of \(t\) will the height of the ball be below 100 feet?

Solve the radical equation to find all real solutions. Check your solutions. $$\sqrt{2 x-1}+2=x$$

In Exercises \(41-48,\) use \(f\) and \(g\) given by the following tables of values. $$\begin{array}{ccccc}x & -1 & 0 & 3 & 6 \\\\\hline f(x) & -2 & 3 & 4 & 2\end{array}$$ $$\begin{array}{ccccc}x & -2 & 1 & 2 & 4 \\\\\hline g(x) & 0 & 6 & -2 & 3\end{array}$$ Evaluate \(g(4)\)

Find \(x+y, x-y, x y,\) and \(x / y\). $$x=2-7 i ; y=11+2 i$$

Use the verbal description to find an algebraic expression for the function. The graph of the function \(g(t)\) is formed by vertically scaling the graph of \(f(t)=|t|\) by a factor of -2 and moving it to the left by 5 units.

Solve the quadratic equation by using the quadratic formula. Find only real solutions. $$x^{2}+x-5=0$$

Decide if each function is odd, even, or neither by using the definitions. $$f(x)=\left(x^{2}+1\right)(x-1)$$

Find the vertex and axis of symmetry of the associated parabola for each quadratic function.Then find at least two additional points on the parabola and sketch the parabola by hand. $$g(x)=-x^{2}+4 x-3$$

Attendance at Broadway shows in New York can be modeled by the quadratic function \(p(t)=0.0489 t^{2}-0.7815 t+10.31,\) where \(t\) is the number of years since 1981 and \(p(t)\) is the attendance in millions. The model is based on data for the years \(1981-2000 .\) For which years was the attendance above 8 million? (Source: The League of American Theaters and Producers, Inc.)

Solve the radical equation to find all real solutions. Check your solutions. $$\sqrt[3]{x+3}=5$$

Find \(x+y, x-y, x y,\) and \(x / y\). $$x=\frac{1}{2}-3 i ; y=\frac{1}{5}+\frac{4}{3} i$$

Use the verbal description to find an algebraic expression for the function. The graph of the function \(k(t)\) is formed by scaling the graph of \(f(t)=\sqrt{t}\) horizontally by a factor of -1 and moving it up 3 units.

Solve the quadratic equation by using the quadratic formula. Find only real solutions. $$-2 x^{2}+2 x+1=0$$

Decide if each function is odd, even, or neither by using the definitions. $$f(x)=\left(x^{2}-3\right)\left(x^{2}-4\right)$$

Find the vertex and axis of symmetry of the associated parabola for each quadratic function.Then find at least two additional points on the parabola and sketch the parabola by hand. $$f(x)=3 x^{2}-12 x+4$$

Solve the radical equation to find all real solutions. Check your solutions. $$\sqrt[3]{5 x-3}=\sqrt[3]{4}$$

In Exercises \(41-48,\) use \(f\) and \(g\) given by the following tables of values. $$\begin{array}{ccccc}x & -1 & 0 & 3 & 6 \\\\\hline f(x) & -2 & 3 & 4 & 2\end{array}$$ $$\begin{array}{ccccc}x & -2 & 1 & 2 & 4 \\\\\hline g(x) & 0 & 6 & -2 & 3\end{array}$$ Evaluate \((f \circ g)(4)\)

Find \(x+y, x-y, x y,\) and \(x / y\). $$x=\frac{1}{3}-2 i ; y=\frac{1}{3}-\frac{2}{5} i$$

Use the verbal description to find an algebraic expression for the function. The graph of the function \(h(x)\) is formed by scaling the graph of \(g(x)=x^{2}\) horizontally by a factor of \(\frac{1}{2}\) and moving it down 4 units.

Solve the quadratic equation by using the quadratic formula. Find only real solutions. $$2 t^{2}+4 t-5=0$$

Find the average rate of change of each ficnetion on the given interval. $$f(x)=-2 x^{2}+5 ; \text { interval: }[-2,-1]$$

Find the vertex and axis of symmetry of the associated parabola for each quadratic function.Then find at least two additional points on the parabola and sketch the parabola by hand. $$h(x)=x^{2}-3 x+5$$

If \(n\) is a positive integer, the sum \(1+2+\cdots+n\) is equal to \(\frac{n(n+1)}{2} .\) For what values of \(n\) will the sum \(1+2+\cdots+n\) be greater than or equal to \(45 ?\)

Solve the radical equation to find all real solutions. Check your solutions. $$\sqrt[4]{x-1}=2$$

In Exercises \(41-48,\) use \(f\) and \(g\) given by the following tables of values. $$\begin{array}{ccccc}x & -1 & 0 & 3 & 6 \\\\\hline f(x) & -2 & 3 & 4 & 2\end{array}$$ $$\begin{array}{ccccc}x & -2 & 1 & 2 & 4 \\\\\hline g(x) & 0 & 6 & -2 & 3\end{array}$$ Evaluate \((g \circ f)(-1)\)

Find \(x+y, x-y, x y,\) and \(x / y\). $$x=-\frac{1}{3}+i \sqrt{5} ; y=-\frac{1}{2}-2 i \sqrt{5}$$

Use the verbal description to find an algebraic expression for the function. The graph of the function \(h(t)\) is formed by scaling the graph of \(f(t)=|t|\) vertically by a factor of \(\frac{1}{2}\) and shifting it up 4 units.

Solve the quadratic equation by using the quadratic formula. Find only real solutions. $$3-x-x^{2}=0$$

Find the average rate of change of each ficnetion on the given interval. $$f(x)=3 x^{2}-1 ; \text { interval: }[2,3]$$

Find the vertex and axis of symmetry of the associated parabola for each quadratic function.Then find at least two additional points on the parabola and sketch the parabola by hand. $$h(x)=-x^{2}+x-2$$

For what value(s) of \(c\) will the inequality \(x^{2}+c>0\) have all real numbers as its solution? Explain.

Solve the radical equation to find all real solutions. Check your solutions. $$\sqrt[4]{2 x+1}=3$$

Find \(x+y, x-y, x y,\) and \(x / y\). $$x=\frac{1}{2}-i \sqrt{3} ; y=\frac{1}{5}+3 i \sqrt{3}$$

Use the verbal description to find an algebraic expression for the function. The graph of the function \(g(x)\) is formed by scaling the graph of \(f(x)=\sqrt{x}\) vertically by a factor of -1 and horizontally by a factor of -1.

Solve the quadratic equation by using the quadratic formula. Find only real solutions. $$-2+t^{2}+t=0$$

Find the average rate of change of each ficnetion on the given interval. $$f(x)=x^{3}+1 ; \text { interval: }[0,2]$$

Find the vertex and axis of symmetry of the associated parabola for each quadratic function.Then find at least two additional points on the parabola and sketch the parabola by hand. $$f(t)=-16 t^{2}+100$$

For what value(s) of \(a\) will the inequality \(a x^{2} \leq 0\) have all real numbers as its solution? Explain.

Solve the radical equation to find all real solutions. Check your solutions. $$\sqrt{2 x+3}-\sqrt{x-2}=2$$

In Exercises \(41-48,\) use \(f\) and \(g\) given by the following tables of values. $$\begin{array}{ccccc}x & -1 & 0 & 3 & 6 \\\\\hline f(x) & -2 & 3 & 4 & 2\end{array}$$ $$\begin{array}{ccccc}x & -2 & 1 & 2 & 4 \\\\\hline g(x) & 0 & 6 & -2 & 3\end{array}$$ Is \((g \circ f)(0)\) defined? Why or why not?

Find \(x+y, x-y, x y,\) and \(x / y\). $$x=-3+i ; y=i+\frac{1}{2}$$

Solve the quadratic equation by using the quadratic formula. Find only real solutions. $$2 x^{2}+x+2=0$$