Question

chapter 2 review

1. the table below shows the relationship between the number of field goals attempted and the number of points scored by one basketball player over a 6 - game period.

field goals attempted (x)	5	6	10	9	7	10
points scored (y)	12	9	14	14	11	15

a) draw a scatter plot for the data. describe the correlation.
b) use (5, 12) and (10, 15) to write a prediction equation in slope - intercept form. then use your prediction equation to predict the number of points scored when 20 field goals are attempted.

1. a) graph the piecewise function.

b) find f(5) and f(0).
f(x)=\

$$\begin{cases}2x + 1&if x\\leq - 1\\ - 4&if - 11\\end{cases}$$

f(5)=-x + 1
f(5)=-5 + 1
f(5)=-4
f(0)=-4

1. write the piecewise function. then state the domain and range in interval notation.
2. you have a summer job that pays time and half for overtime. that means, if you work more than 40 hours a week, your hourly wage for the extra hours is 1.5 times your normal rate of $7 per hour.

a) write a piecewise function to model your weekly pay, p, in terms of the number of hours worked, x.
b) how much will you get paid if you work 30 hrs?
c) how much will you get paid if you work 47 hrs?

1. sweets bakery charges $12 for each pie and $15 for each cake. yesterday, the bakery took in no more than $360 for sales of pies and cakes. write an inequality to represent the situation, where p is the number of pies sold and c is the number of cakes sold. then graph the inequality.

Explanation:

1.

a)

To draw a scatter - plot:

1. On the x - axis, mark the number of field goals attempted and on the y - axis, mark the number of points scored.
2. Plot the points \((5,12)\), \((6,9)\), \((10,14)\), \((9,14)\), \((7,11)\), \((10,15)\).
3. The correlation appears to be positive. As the number of field goals attempted increases, the number of points scored also tends to increase.

b)

The slope - intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.

Step1: Calculate the slope \(m\)

The slope \(m\) between two points \((x_1,y_1)=(8,12)\) and \((x_2,y_2)=(10,15)\) is given by the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

$$m=\frac{15 - 12}{10 - 8}=\frac{3}{2}=1.5$$

Step2: Find the y - intercept \(b\)

We use the point - slope form \(y - y_1=m(x - x_1)\) with the point \((8,12)\) and \(m = 1.5\).

$$y-12 = 1.5(x - 8)$$

$$y-12=1.5x-12$$

$$y = 1.5x$$

To predict the number of points scored when \(x = 20\) field goals are attempted:
Substitute \(x = 20\) into the equation \(y = 1.5x\).

$$y=1.5\times20=30$$

2.

a)

To graph the piece - wise function \(f(x)=

$$\begin{cases}2x + 1, &x\leq - 1\\-4, &-1\lt x\leq1\\-x + 1, &x\gt1\end{cases}$$

\):

1. For \(y = 2x+1\) when \(x\leq - 1\):

• When \(x=-1\), \(y=2\times(-1)+1=-1\). The line \(y = 2x + 1\) has a slope of 2 and a y - intercept of 1. We draw the part of this line for \(x\leq - 1\).

1. For \(y=-4\) when \(-1\lt x\leq1\), we draw a horizontal line at \(y = - 4\) from \(x=-1\) (open circle) to \(x = 1\) (closed circle).
2. For \(y=-x + 1\) when \(x\gt1\):

• When \(x = 1\), \(y=-1 + 1=0\). The line \(y=-x + 1\) has a slope of \(-1\) and a y - intercept of 1. We draw the part of this line for \(x\gt1\) (open circle at \(x = 1\)).

b)

To find \(f(5)\) and \(f(0)\):

1. Since \(x = 5\gt1\), we use \(f(x)=-x + 1\).

$$f(5)=-5 + 1=-4$$

1. Since \(0\) satisfies \(-1\lt0\leq1\), we use \(f(x)=-4\).

$$f(0)=-4$$

3.

The piece - wise function from the graph:

$$f(x)=
LATEXBLOCK1
$$

The domain is \((-\infty,\infty)\) (all real numbers) in interval notation.
The range is \((-\infty,\infty)\) (all real numbers) in interval notation.

4.

a)

The piece - wise function for the weekly pay \(P(x)\):

$$P(x)=
LATEXBLOCK2
$$

$$P(x)=
LATEXBLOCK3
$$

$$P(x)=
LATEXBLOCK4
$$

b)

If \(x = 30\) (since \(30\leq40\)), we use \(P(x)=7x\).

$$P(30)=7\times30 = 210$$

c)

If \(x = 47\) (since \(47\gt40\)), we use \(P(x)=10.5x-140\).

$$P(47)=10.5\times47-140=493.5-140 = 353.5$$

5.

The inequality representing the situation:
The cost of pies is \(12p\) and the cost of cakes is \(15c\). The total sales are no more than \(360\). So the inequality is \(12p+15c\leq360\).
To graph the inequality:

1. First, rewrite it in slope - intercept form for \(c\):

$$15c\leq - 12p+360$$

$$c\leq-\frac{4}{5}p + 24$$

1. The boundary line is \(c=-\frac{4}{5}p + 24\).

• When \(p = 0\), \(c = 24\).
• When \(c = 0\), \(p = 30\).

1. Since the inequality is \(\leq\), we draw a solid line and shade the region below the line (including the line itself) in the first - quadrant (because \(p\geq0\) and \(c\geq0\) as they represent the number of pies and cakes sold).

Answer:

1. a) Scatter - plot shows positive correlation.

b) Prediction equation is \(y = 1.5x\), and the predicted number of points when \(x = 20\) is 30.

1. a) Graph as described above.

b) \(f(5)=-4\), \(f(0)=-4\).

1. Piece - wise function: \(f(x)=

$$\begin{cases}-2x-1, &x\lt - 1\\1, &-1\leq x\lt1\\x - 1, &x\geq1\end{cases}$$

\), Domain: \((-\infty,\infty)\), Range: \((-\infty,\infty)\).

1. a) \(P(x)=

$$\begin{cases}7x, &0\leq x\leq40\\10.5x-140, &x\gt40\end{cases}$$

\)
b) 210
c) 353.5

1. Inequality: \(12p + 15c\leq360\), graph as described above.