Chapter 1.2, Problem 5E

(a)

To determine

To find: An equation of the family of linear function with slope 2 and sketch the members of the family on the graph.

Answer to Problem 5E

An equation for the family of linear function with slope 2 is $f\left(x\right)=2x+c$.

Explanation of Solution

The general equation of a linear function is of the form $y=mx+c$ with slope m and y-intercept c.

Given that, the slope (m) is 2.

Thus, the linear function of slope 2 is $y=2x+c$ where c is any real number.

Obtain the equations for the various c values.

If $c=1$, then $y=2x+1$.

If $c=0$, then $y=2x$.

If $c=-1$, then $y=2x-1$.

If $c=-2$, then $y=2x-2$.

Draw the graph of the above equations as shown below in Figure 1.

From Figure 1, it is noticed that all the members of the family of linear function $f\left(x\right)=2x+c$ having the common slope, m = 2.

(b)

To determine

To find: The equation of the family of linear function satisfying the condition that $f\left(2\right)=1$ and sketch the members of the family on the graph.

Answer to Problem 5E

The equation for the family of linear function with the condition $f\left(2\right)=1$ is $f\left(x\right)=mx+\left(1-2m\right)$.

Explanation of Solution

The general equation of a linear function is of the form $y=mx+c$ with slope m and y-intercept c.

Since the equation satisfies the condition $f\left(2\right)=1$, the point $\left(2,1\right)$ lie on the graph.

Substitute the point $\left(x,y\right)=\left(2,1\right)$ in point-slope form and obtain the equation.

$\begin{array}{l}y-y_1=m\left(x-x_1\right)\left[\text{Point-slope form}\right]\\ y-1=m\left(x-2\right)\\ y=mx-2m+1\\ y=mx+\left(1-2m\right)\end{array}$

Thus, the required linear function is $y=mx+\left(1-2m\right)$.

Obtain the equations for the various m values.

If $m=2$, then $y=2x-3$.

If $m=1$, then $y=x-1$.

If $m=0$, then $y=1$.

If $m=-1$, then $y=-x+3$.

Draw the graph of the above equations as shown below in Figure 2.

From Figure 2, it is noticed that all the members of the family of linear function $y=mx+\left(1-2m\right)$ passing through the common point (2, 1).

(c)

To determine

To find: The common function belongs to both the families of part (a) and part (b).

Answer to Problem 5E

The common function belongs to both the families is $y=2x-3$.

Explanation of Solution

The function that belongs to both families, must satisfies the conditions that the slope, m = 2 and $f\left(2\right)=1$.

Substitute m = 2 and the point $\left(x,y\right)=\left(2,1\right)$ in point-slope form and obtain the equation of common function.

$\begin{array}{l}y-y_1=m\left(x-x_1\right)\left[\text{Point-slope form}\right]\\ y-1=2\left(x-2\right)\\ y=2x-4+1\\ y=2x-3\end{array}$

Thus, the required function is $y=2x-3$.

The graph of the function $y=2x-3$ is shown below in Figure 3.

From Figure 3, it is observed that the conditions of both parts (a) and (b) are satisfied. That is, the graph of slope 2 and passing through the point (2, 1).