Chapter 3

Solve each problem. The heart rate of an athlete while weight training is recorded for 4 minutes. The table lists the heart rate after \(x\) minutes. $$\begin{array}{|l|c|c|c|c|c|} \hline \text { Time (min) } & 0 & 1 & 2 & 3 & 4 \\ \hline \begin{array}{l} \text { Heart rate } \\ \text { (bpm) } \end{array} & 84 & 111 & 120 & 110 & 85 \\ \hline \end{array}$$ (a) Explain why the data are not linear. (b) Find a quadratic function \(f\) that models the data. Use \((2,120)\) as the vertex of the parabola. (c) What is the domain of the function?

Multiply or divide as indicated. Simplify each answer. $$(-2+3 i)-(-4+3 i)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$\frac{2}{3} x^{2}+\frac{1}{4} x=3$$

Solve each problem. The table shows a person's heart rate during the first 4 minutes after exercise has stopped. $$\begin{array}{|l|c|c|c|} \hline \text { Time (min) } & 0 & 2 & 4 \\ \hline \text { Heart rate (bpm) } & 154 & 106 & 90 \\ \hline \end{array}$$ (a) Find a formula \(f(x)=a(x-h)^{2}+k\) that models the data, where \(x\) represents time and \(0 \leq x \leq 4 .\) Use \((4,90)\) as the vertex. (b) Evaluate \(f(1)\) and interpret the result. (c) Estimate the times when the heart rate was from 115 to 125 beats per minute.

Solve each equation. For equations with real solutions, support your answers graphically. $$(3-x)^{2}=25$$

Solve each equation. For equations with real solutions, support your answers graphically. $$(2+x)^{2}=49$$

Multiply or divide as indicated. Simplify each answer. $$(9-5 i)-(3 i-6)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$2 x^{2}-4 x=1$$

Multiply or divide as indicated. Simplify each answer. $$(2-5 i)-(3+4 i)-(-2+i)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$3 x^{2}-6 x=4$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=-1-x$$

Multiply or divide as indicated. Simplify each answer. $$(-6+5 i)+(4-4 i)+(2-i)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=-3-3 x$$

Multiply or divide as indicated. Simplify each answer. $$(7+9 i)+(1-2 i)+(-8-7 i)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$4 x^{2}-20 x+25=0$$

Find the equation of the quadratic function satisfying the given conditions. (Hint: Determine values of \(a\), \(h,\) and \(k\) that satisfy \(P(x)=a(x-h)^{2}+k .\) ) Express your answer in the form \(P(x)=a x^{2}+b x+c\). Use your calculator to support your results. Vertex \((-1,-4) ;\) through \((5,104)\)

Multiply as indicated. Write each product in standand form. $$(2+i)(3-2 i)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$9 x^{2}+12 x+4=0$$

Find the equation of the quadratic function satisfying the given conditions. (Hint: Determine values of \(a\), \(h,\) and \(k\) that satisfy \(P(x)=a(x-h)^{2}+k .\) ) Express your answer in the form \(P(x)=a x^{2}+b x+c\). Use your calculator to support your results. Vertex \((-2,-3) ;\) through \((0,-19)\)

Multiply as indicated. Write each product in standand form. $$(-2+3 i)(4-2 i)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$-3 x^{2}+4 x+4=0$$

Find the equation of the quadratic function satisfying the given conditions. (Hint: Determine values of \(a\), \(h,\) and \(k\) that satisfy \(P(x)=a(x-h)^{2}+k .\) ) Express your answer in the form \(P(x)=a x^{2}+b x+c\). Use your calculator to support your results. Vertex \((8,3) ;\) through \((10,5)\)

Multiply as indicated. Write each product in standand form. $$(2+4 i)(-1+3 i)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$-5 x^{2}+28 x+12=0$$

Find the equation of the quadratic function satisfying the given conditions. (Hint: Determine values of \(a\), \(h,\) and \(k\) that satisfy \(P(x)=a(x-h)^{2}+k .\) ) Express your answer in the form \(P(x)=a x^{2}+b x+c\). Use your calculator to support your results. Vertex \((-6,-12) ;\) through \((6,24)\)

Multiply as indicated. Write each product in standand form. $$(1+3 i)(2-5 i)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$(x+5)(x-6)=(2 x-1)(x-4)$$

Find the equation of the quadratic function satisfying the given conditions. (Hint: Determine values of \(a\), \(h,\) and \(k\) that satisfy \(P(x)=a(x-h)^{2}+k .\) ) Express your answer in the form \(P(x)=a x^{2}+b x+c\). Use your calculator to support your results. Vertex \((-4,-2) ;\) through \((2,-26)\)

Multiply as indicated. Write each product in standand form. $$(-3+2 i)^{2}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$(x+2)(3 x-4)=(x+5)(2 x-5)$$

Find the equation of the quadratic function satisfying the given conditions. (Hint: Determine values of \(a\), \(h,\) and \(k\) that satisfy \(P(x)=a(x-h)^{2}+k .\) ) Express your answer in the form \(P(x)=a x^{2}+b x+c\). Use your calculator to support your results. Vertex \((5,6) ;\) through \((1,-6)\)

Multiply as indicated. Write each product in standand form. $$(2+i)^{2}$$

Solve each quadratic equation by completing the square. $$x^{2}-2 x=2$$

A rocket is launched upward from ground level with an initial velocity of 90 feet per second. Let \(t\) represent the amount of time elapsed after it is launched. (a) Explain why \(t\) cannot be a negative number in this situation. (b) Explain why \(s_{0}=0\) in this problem. (c) Give the function \(s\) that describes the height of the rocket as a function of \(t\) (d) How high will the rocket be 1.5 seconds after it is launched? (e) What is the maximum height attained by the rocket? After how many seconds will this happen? Determine the answer analytically and graphically. (f) After how many seconds will the rocket hit the ground? Determine the answer graphically.

Multiply as indicated. Write each product in standand form. $$(3+i)(-3-i)$$

Solve each quadratic equation by completing the square. $$x^{2}+2 x=4$$

A toy rocket is launched from the top of a building 50 feet tall at an initial velocity of 200 feet per second. Let \(t\) represent the amount of time elapsed after the launch. (a) Express the height \(s\) as a function of the time \(t\) (b) Determine both analytically and graphically the time at which the rocket reaches its highest point. How high will it be at that time? (c) For what time interval will the rocket be more than 300 feet above ground level? Determine the answer graphically, and give times to the nearest tenth of a second. (d) After how many seconds will the rocket hit the ground? Determine the answer graphically.

Multiply as indicated. Write each product in standand form. $$(-5-i)(5+i)$$

Solve each quadratic equation by completing the square. $$2 x^{2}+6 x-3=0$$

Multiply as indicated. Write each product in standand form. $$(2+3 i)(2-3 i)$$

Solve each quadratic equation by completing the square. $$3 x^{2}-3 x-1=0$$

An astronaut on the moon throws a baseball upward. The astronaut is 6 feet, 6 inches tall and the initial velocity of the ball is 30 feet per second. The height of the ball is approximated by the function $$s(t)=-2.7 t^{2}+30 t+6.5$$ where \(t\) is the number of seconds after the ball was thrown. (a) After how many seconds is the ball 12 feet above the moon's surface? (b) How many seconds after it is thrown will the ball return to the surface? (c) The ball will never reach a height of 100 feet. How can this be determined analytically?

Multiply as indicated. Write each product in standand form. $$(6-4 i)(6+4 i)$$

Solve each quadratic equation by completing the square. $$x(x-1)=3$$

Multiply as indicated. Write each product in standand form. $$(\sqrt{6}+i)(\sqrt{6}-i)$$

Solve each quadratic equation by completing the square. $$x(x-2)=4$$

Multiply as indicated. Write each product in standand form. $$(\sqrt{2}-4 i)(\sqrt{2}+4 i)$$

Solve each quadratic equation by completing the square. $$2 x^{2}-x=-3$$

Multiply as indicated. Write each product in standand form. $$i(3-4 i)(3+4 i)$$

Solve each quadratic equation by completing the square. $$2 x(2 x-5)=2$$