Chapter 3

Determine whether each statement is true or false. If it is false, tell why. Every real number is a complex number.

For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=3 x^{2}+4 x-1$$

Solve each problem. A farmer wishes to enclose a rectangular region bordering a barn with fencing, as shown in the diagram. Suppose that \(x\) represents the length of each of the three parallel pieces of fencing. She has 600 feet of fencing available. (a) What is the length of the remaining piece of fencing in terms of \(x ?\) (b) Determine a function \(s t\) that represents the total area of the enclosed region. Give any restrictions on \(x .\) (c) What dimensions for the total enclosed region would give an area of \(22,500\) square feet? Determine the answer analytically. (d) Use a graph to find the maximum area that can be enclosed.

Solve each equation. For equations with real solutions, support your answers graphically. $$2 x^{2}=48$$

Determine whether each statement is true or false. If it is false, tell why. No real number is a pure imaginary number.

For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=4 x^{2}+3 x-1$$

Solve each problem. Hitting a Baseball A baseball is hit so that its height in feet after \(t\) seconds is given by $$s(t)=-16 t^{2}+44 t+4$$ (a) How high is the baseball after 1 second? (b) Find the maximum height of the baseball.

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=-16$$

Determine whether each statement is true or false. If it is false, tell why. Every pure imaginary number is a complex number.

Match each function in Column I with the description of the parabola that is its graph in Column II. \(\mathbf{I}\) (a) \(f(x)=(x-4)^{2}-2\) (b) \(f(x)=(x-2)^{2}-4\) (c) \(f(x)=-(x-4)^{2}-2\) (d) \(f(x)=-(x-2)^{2}-4\) \(\mathbf{I I}\) A. Vertex \((2,-4),\) opens downward B. Vertex \((2,-4),\) opens upward C. Vertex \((4,-2),\) opens downward D. Vertex \((4,-2),\) opens upward

Solve each problem. A golf ball is hit so that its height \(h\) in feet after \(t\) seconds is given by $$h(t)=-16 t^{2}+60 t$$ (a) What is the initial height of the golf ball? (b) How high is the golf ball after 1.5 seconds? (c) Find the maximum height of the golf ball.

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=-100$$

Determine whether each statement is true or false. If it is false, tell why. Every pure imaginary number is a complex number.

Match each function in Column I with the description of the parabola that is its graph in Column II, assuming \(a>0, h>0,\) and \(k>0\). \(\mathbf{I}\) (a) \(f(x)=-a(x+h)^{2}+k\) (b) \(f(x)=a(x-h)^{2}+k\) (c) \(f(x)=a(x+h)^{2}+k\) (d) \(f(x)=-a(x-h)^{2}+k\) \(\mathbf{II}\) A. Vertex in quadrant I, two \(x\)-intercepts B. Vertex in quadrant I, no \(x\)-intercepts C. Vertex in quadrant II, two \(x\)-intercepts D. Vertex in quadrant II, no \(x\)-intercepts

Solve each problem. 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.

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=-18$$

Determine whether each statement is true or false. If it is false, tell why. There is no real number that is a complex number.

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=-32$$

Solve each problem. A cylindrical aluminum can is being constructed to have a height \(h\) of 4 inches. If the can is to have a volume of 28 cubic inches, approximate its radius \(r\) (Hint: \(\left.V=\pi r^{2} h .\right)\)

Determine whether each statement is true or false. If it is false, tell why. A complex number might not be a pure imaginary number.

Solve each equation. For equations with real solutions, support your answers graphically. $$(3 x-1)^{2}=12$$

Solve each problem. A piece of cardboard is twice as long as it is wide. It is to be made into a box with an open top by cutting 2 -inch squares from each corner and folding up the sides. Let \(x\) represent the width of the original piece of cardboard. (a) Represent the length of the original piece of cardboard in terms of \(x\) (b) What will be the dimensions of the bottom rectangular base of the box? Give the restrictions on \(x .\) (c) Determine a function \(V\) that represents the volume of the box in terms of \(x .\) (d) For what dimensions of the bottom of the box will the volume be 320 cubic inches? Determine analytically and support graphically. (e) Determine graphically (to the nearest tenth of an inch) the values of \(x\) if the box is to have a volume between 400 and 500 cubic inches.

Write each number in simplest form, without a negative radicand. $$\sqrt{-100}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$(4 x+1)^{2}=20$$

Write each number in simplest form, without a negative radicand. $$ \sqrt{-169} $$

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

Solve each problem. A can of peas and carrots has surface area 54.19 square inches. Its height is 4.25 inches. What is the radius of the circular top, to the nearest tenth of an inch? See the figure. (Hint: The surface area consists of the circular top and bottom and a rectangle that represents the side cut open vertically and unrolled.)

Write each number in simplest form, without a negative radicand. $$ -\sqrt{-400} $$

Solve each equation. For equations with real solutions, support your answers graphically. $$(-2 x+5)^{2}=-8$$

Solve each problem. The volume of a 10 -ounce box of cereal is 182.742 cubic inches. The width of the box is 3.1875 inches less than the length, and its depth is 2.3125 inches. Find the length and width of the box to the nearest thousandth of an inch.

Determine whether each statement is true or false. If it is false, tell why. $$-\sqrt{-225}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=2 x+24$$

Solve each problem. A square lawn has area 800 square feet. A sprinkler placed at the center of the lawn sprays water in a circular pattern that just covers the lawn. What is the radius of the circle?

Write each number in simplest form, without a negative radicand. $$ -\sqrt{-39} $$

For quadratic function, (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=x^{2}-10 x+21$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=3 x+18$$

Solve each problem. A kite is flying on 50 feet of string. How high is it above the ground if its height is 10 feet more than the horizontal distance from the person flying it? Assume that the string is being held at ground level.

Determine whether each statement is true or false. If it is false, tell why. $$-\sqrt{-95}$$

For quadratic function, (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=x^{2}-2 x+3$$

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

Solve each problem. A boat with a rope attached at water level is being pulled into a dock. When the boat is 12 feet from the dock, the length of the rope is 3 feet more than twice the height of the dock above the water. Find the height of the dock.

Determine whether each statement is true or false. If it is false, tell why. $$5+\sqrt{-4}$$

For quadratic function, (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=-x^{2}+4 x-2$$

Solve each equation. For equations with real solutions, support your answers graphically. $$5 x^{2}+3 x=0$$

Solve each problem. A raised wooden walkway is being constructed through a wetland. The walkway will have the shape of a right triangle with one leg 700 yards longer than the other and the hypotenuse 100 yards longer than the longer leg. Find the total length of the walkway.

Determine whether each statement is true or false. If it is false, tell why. $$-7+\sqrt{-100}$$

For quadratic function, (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=-x^{2}+2 x+1$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x(14 x+1)=3$$

Solve each problem. A building is 2 feet from a 9 -foot fence that surrounds the property. A worker wants to wash a window in the building 13 feet from the ground. He plans to place a ladder over the fence so that it rests against the building. He decides he should place the ladder at least 8 feet from the fence for stability. To the nearest foot, how long a ladder will he need?

For quadratic function, (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=2 x^{2}-4 x+5$$