Chapter 7.1, Problem 8LC

Solution

a.

To Identify: Whether the given function is linear, quadratic, or exponential.

The equation $y=3{\left(x+1\right)}^2$ is a quadratic equation.

Given:

  $y=3{\left(x+1\right)}^2$

Explanation:

The given function is $y=3{\left(x+1\right)}^2$

Rewrite ${\left(x+1\right)}^2$ as,

  $3\left(\left(x+1\right)\left(x+1\right)\right)$

Expand $\left(x+1\right)\left(x+1\right)$ using the FOIL Method:

  $3\left(x\cdot x+x\cdot 1+1x+1\cdot 1\right)$

Simplify and combine like terms:

  $3\left(x^2+2x+1\right)$

Remove the parenthesis:

  $3x^2+6x+3$

Since, the equation is in the form of $a{x}^2+bx+c=0$, it is a quadratic equation.

b.

To Identify: Whether the given function is linear, quadratic, or exponential.

The equation $y=4{\left(3\right)}^x$ is an exponential equation.

Given:

  $y=4{\left(3\right)}^x$

Explanation:

The given function is $y=4{\left(3\right)}^x$

The given function is in the form $y=a{b}^x$ where $a\ne 0$ with $b>0$ and $b\ne 1$

Hence the given function is an exponential function.

c.

To Identify: Whether the given function is linear, quadratic, or exponential.

The equation $y=2x+5$ is a linear equation.

Given:

  $y=2x+5$

Explanation:

The given function is $y=2x+5$

The linear function is in the form of $f\left(x\right)=ax+b$ where $a$ and $b$ are constants and $a$ is not equal to zero. Furthermore, the highest degree of the equation should be $1$ .

In the equation $y=2x+5$, $a=2$ and $b=5$ which are constants. And $a$ is not equal to zero. Also, the highest degree of the equation is also $1$ .

Hence the given function is a linear function.

d.

To Identify: Whether the given function is linear, quadratic, or exponential.

The equation $y=4{\left(0.2\right)}^x+1$ is a linear equation.

Given:

  $y=4{\left(0.2\right)}^x+1$

Explanation:

The given function is $y=4{\left(0.2\right)}^x+1$

It is in the form $y=a{b}^x$ where $a\ne 0$ with $b>0$ and $b\ne 1$

Hence the given function is an exponential function.