91Ó°ÊÓ

Explain why the inequality \(x^{2}-x+1<0\) has the empty set as the solution set.

Suppose that the quantity supplied \(S\) and the quantity demanded \(D\) of hot dogs at a baseball game are given by the following functions:$$\begin{array}{l}S(p)=-2000+3000 p \\\D(p)=10,000-1000 p\end{array}$$ where \(p\) is the price of a hot dog. (a) Find the equilibrium price for hot dogs at the baseball game. What is the equilibrium quantity? (b) Determine the prices for which quantity demanded is less than quantity supplied. (c) What do you think will eventually happen to the price of hot dogs if quantity demanded is less than quantity supplied?

The simplest cost function \(C\) is a linear cost function, \(C(x)=m x+b,\) where the \(y\) -intercept \(b\) represents the fixed costs of operating a business and the slope \(m\) represents the cost of each item produced. Suppose that a small bicycle manufacturer has daily fixed costs of \(\$ 1800,\) and each bicycle costs \(\$ 90\) to manufacture. (a) Write a linear model that expresses the cost \(C\) of manufacturing \(x\) bicycles in a day. (b) Graph the model. (c) What is the cost of manufacturing 14 bicycles in a day? (d) How many bicycles could be manufactured for \(\$ 3780 ?\)

(a) find the vertex and the axis of symmetry of each quadratic function, and determine whether the graph is concave up or concave down. (b) Find the y-intercept and the \(x\) -intercepts, if any. (c) Use parts (a) and (b) to graph the function. (d) Find the domain and the range of the quadratic function. (e) Determine where the quadratic function is increasing and where it is decreasing. (f) Determine where \(f(x)>0\) and where \(f(x)<0\) \(f(x)=x^{2}+6 x+9\)

According to Hooke's Law, a linear relationship exists between the distance that a spring stretches and the force stretching it. Suppose a weight of 0.5 kilograms causes a spring to stretch 2.75 centimeters and a weight of 1.2 kilograms causes the same spring to stretch 6.6 centimeters. (a) Find a linear model that relates the distance \(d\) of the stretch and the weight \(w\). (b) What stretch is caused by a weight of 2.4 kilograms? (c) What weight causes a stretch of 19.8 centimeters?

(a) find the vertex and the axis of symmetry of each quadratic function, and determine whether the graph is concave up or concave down. (b) Find the y-intercept and the \(x\) -intercepts, if any. (c) Use parts (a) and (b) to graph the function. (d) Find the domain and the range of the quadratic function. (e) Determine where the quadratic function is increasing and where it is decreasing. (f) Determine where \(f(x)>0\) and where \(f(x)<0\) \(f(x)=4 x^{2}-2 x+1\)

Temperature Conversion The linear function \(F(C)=\frac{9}{5} C+32\) converts degrees Celsius to degrees Fahrenheit, and the linear function \(R(F)=F+459.67\) converts degrees Fahrenheit to degrees Rankine. Find a linear function that converts degrees Rankine to degrees Celsius.

(a) find the vertex and the axis of symmetry of each quadratic function, and determine whether the graph is concave up or concave down. (b) Find the y-intercept and the \(x\) -intercepts, if any. (c) Use parts (a) and (b) to graph the function. (d) Find the domain and the range of the quadratic function. (e) Determine where the quadratic function is increasing and where it is decreasing. (f) Determine where \(f(x)>0\) and where \(f(x)<0\) \(f(x)=-2 x^{2}+2 x-3\)

The following data represent the various combinations of soda and hot dogs that Yolanda can buy at a baseball game with \(\$60$$\begin{array}{|cc|}\hline \text { Soda, } s & \text { Hot Dogs, } h \\\\\hline 20 & 0 \\\15 & 3 \\\10 & 6 \\\5 & 9 \\\\\hline\end{array}$$ (a) Plot the ordered pairs \)(s, h)\( in a Cartesian plane. (b) Show that the number \)h\( of hot dogs purchased is a linear function of the number \)s\( of sodas purchased. (c) Determine the linear function that describes the relation between \)s\( and \)h$ (d) What is the domain of the linear function? (e) Graph the linear function in the Cartesian plane drawn in part (a). (f) Interpret the slope. (g) Interpret the intercepts.

(a) find the vertex and the axis of symmetry of each quadratic function, and determine whether the graph is concave up or concave down. (b) Find the y-intercept and the \(x\) -intercepts, if any. (c) Use parts (a) and (b) to graph the function. (d) Find the domain and the range of the quadratic function. (e) Determine where the quadratic function is increasing and where it is decreasing. (f) Determine where \(f(x)>0\) and where \(f(x)<0\) \(f(x)=2 x^{2}+5 x+3\)