Chapter 2

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$5 x^{2}-2 x+3=0$$

For function of the form \(f(x)=\) \(a x^{2}+b x+c,\) find the discriminant, \(b^{2}-4 a c,\) and use it to determine the number of \(x\) -intercepts of the graph of \(f .\) Also determine the number of real solutions of the equation \(f(x)=0.\) $$f(x)=x^{2}-2 x-1$$

Applications In this set of exercises, you will use properties of functions to study real-world problems. Stamp Collecting The value of a commemorative stamp \(t\) years after purchase appreciates according to the function \(v(t)=0.37+0.05 t,\) where \(v(t)\) is in dollars. Find the average rate of change of the value of the stamp on the interval \([0,4],\) and interpret it.

Graph each quadratic function by finding a suitable viewing window with the help of the TABLE feature of a graphing utility. Also find the vertex of the associated parabola using the graphing utility. $$h(x)=x^{2}+5 x-20$$

In Exercises \(49-66,\) let \(f(x)=x^{2}+x, g(x)=\sqrt{x},\) and \(h(x)=-3 x\) Evaluate each of the following. $$(h \circ f)\left(\frac{3}{2}\right)$$

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$-7 x^{2}+2 x-1=0$$

For function of the form \(f(x)=\) \(a x^{2}+b x+c,\) find the discriminant, \(b^{2}-4 a c,\) and use it to determine the number of \(x\) -intercepts of the graph of \(f .\) Also determine the number of real solutions of the equation \(f(x)=0.\) $$f(x)=-x^{2}+x+3$$

Applications In this set of exercises, you will use properties of functions to study real-world problems. Commerce The following table lists the annual sales of CDs by a small music store for selected years. $$ \begin{array}{c|c} \text { Year } & \text { Number of Units Sold } \\ 2002 & 10,000 \\ 2005 & 30,000 \\ 2006 & 33,000 \end{array} $$ Find the average rate of change in sales from 2002 to 2005. Also find the average rate of change in sales from 2005 to \(2006 .\) Does the average rate of change stay the same for both intervals? Why would a linear function \(n o t\) be useful for modeling these sales figures?

Graph each quadratic function by finding a suitable viewing window with the help of the TABLE feature of a graphing utility. Also find the vertex of the associated parabola using the graphing utility. $$s(t)=-16 t^{2}+40 t+120$$

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=-x^{2}+1 ; g(x)=x+1$$

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$5 x^{2}=-2 x-3$$

For function of the form \(f(x)=\) \(a x^{2}+b x+c,\) find the discriminant, \(b^{2}-4 a c,\) and use it to determine the number of \(x\) -intercepts of the graph of \(f .\) Also determine the number of real solutions of the equation \(f(x)=0.\) $$f(x)=2 x^{2}+x+1$$

Graph the function \(f(t)=t^{2}-4\) in a decimal window. Using your graph, determine the values of \(t\) for which \(f(t) \geq 0\).

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=2 x+5 ; g(x)=3 x^{2}$$

For function of the form \(f(x)=\) \(a x^{2}+b x+c,\) find the discriminant, \(b^{2}-4 a c,\) and use it to determine the number of \(x\)-intercepts of the graph of \(f .\) Also determine the number of real solutions of the equation \(f(x)=0.\) $$f(x)=3 x^{2}-4 x+4$$

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$7 x^{2}=-x-1$$

Suppose \(f\) is constant on an interval \([a, b] .\) Show that the average rate of change of \(f\) on \([a, b]\) is zero.

Suppose the vertex of the parabola associated with a certain quadratic function is \((2,1),\) and another point on this parabola is (3,-1) (a) Find the equation of the axis of symmetry of the parabola. (b) Use symmetry to find a third point on the parabola. (c) Sketch the parabola.

This set of exercises will draw on the ideas presented in this section and your general math background. Explain what is wrong with the following steps for solving a radical equation. $$\begin{aligned}\sqrt{x+1}-2 &=0 \\\\(x+1)+4 &=0 \\\x &=-5\end{aligned}$$

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=4 x-1 ; g(x)=\frac{x+1}{4}$$

For function of the form \(f(x)=\) \(a x^{2}+b x+c,\) find the discriminant, \(b^{2}-4 a c,\) and use it to determine the number of \(x\)-intercepts of the graph of \(f .\) Also determine the number of real solutions of the equation \(f(x)=0.\) $$f(x)=x^{2}+2 x+1$$

Use a graphing utility to solve the problem. Graph \(f(x)=|x+3.5|\) and \(g(x)=|x|+3.5 .\) Describe each graph in terms of transformations of the graph of \(h(x)=|x|\).

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$-3 x^{2}+8 x=16$$

If the average rate of change of a function on an interval is zero, docs that mean the function is constant on that interval?

Examine the following table of values for a quadratic function \(f\) $$\begin{array}{rr} x & f(x) \\ -2 & 3 \\ -1 & 0 \\ 0 & -1 \\ 1 & 0 \\ 2 & 3 \end{array}$$ (a) What is the equation of the axis of symmetry of the associated parabola? Justify your answer. (b) Find the minimum or maximum value of the function and the value of \(x\) at which it occurs. (c) Sketch a graph of the function from the values given in the table, and find an expression for the function.

For function of the form \(f(x)=\) \(a x^{2}+b x+c,\) find the discriminant, \(b^{2}-4 a c,\) and use it to determine the number of \(x\)-intercepts of the graph of \(f .\) Also determine the number of real solutions of the equation \(f(x)=0.\) $$f(x)=-x^{2}+4 x-4$$

This set of exercises will draw on the ideas presented in this section and your general math background. Without doing any calculations, explain why $$\sqrt{x+1}=-2$$ does not have a solution.

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=2 x+3 ; g(x)=\frac{x-3}{2}$$

Use a graphing utility to solve the problem. Graph \(f(x)=(x-4.5)^{2}\) and \(g(x)=x^{2}+4.5 .\) Describe each graph in terms of transformations of the graph of \(h(x)=x^{2}\).

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$2 t^{2}+8 t=-9$$

If the average rate of change of a function on an interval is zero, does that mean the function is constant on that interval?

Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$G(x)=-6 x+x^{2}+5$$

Let \(g(s)=-2 s^{2}+b s .\) Find the value of \(b\) such that the vertex of the parabola associated with this function is (1,2)

This set of exercises will draw on the ideas presented in this section and your general math background. How many zeros, real and nonreal, does the function \(f(x)=x^{4}-1\) have? How many \(x\) -intercepts does the graph of \(f\) have?

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=3 x^{2}+4 x ; g(x)=x+2$$

Use a graphing utility to solve the problem. If \(f(x)=\sqrt{x}, \operatorname{graph} f(x)\) and \(f(x-4.5)\) in the same viewing window. What is the relationship between the two graphs?

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$-4 t^{2}+t-\frac{1}{2}=0$$

Let \(f\) be decreasing on an interval \((a, b) .\) Show that the average rate of change of \(f\) on \([c, d]\) is negative, where \(a

Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$h(t)=-5 t+3-t^{2}$$

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=-2 x+1 ; g(x)=2 x^{2}-5 x$$

Use a graphing utility to solve the problem. If \(f(x)=|x|,\) graph \(f(x)\) and \(f(0.3 x)\) in the same viewing window. What is the relationship between the two graphs?

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$-6 t^{2}+2 t-\frac{1}{3}=0$$

Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$F(s)=-2 s^{2}+3 s+1$$

A rectangular fence is being constructed around a new play area at a local elementary school. If the school has 2000 feet of fencing available for the project, what is the maximum area that can be enclosed for the new play area?

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=\frac{1}{x} ; g(x)=2 x+5$$

Use a graphing utility to solve the problem. If \(f(x)=|x|,\) graph \(-2 f(x)\) and \(f(-2 x)\) in the same viewing window. Are the graphs the same? Explain.

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$\frac{2}{3} x^{2}+x=-1$$

Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$g(t)=3 t^{2}-6 t-\frac{3}{4}$$

A rectangular garden plot is to be enclosed with a fence on three of its sides and a brick wall on the fourth side. If 100 feet of fencing material is available, what dimensions will yield the maximum area? 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. Here, \(t\) is in seconds and \(h(t)\) is in feet. (a) Sketch a graph of \(h\) (b) When will the ball reach its maximum height, and what is the maximum height?

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=3 x+1 ; g(x)=\frac{2}{x}$$