Chapter 2.5, Problem 52AYU

In Problems 39-62, graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Stan with the graph of the basic function (for example, $y\text{ = }{x}^2$) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function.

52. $f\left(x\right)\text{ = 3}{\left(x\text{ - 2}\right)}^2\text{ + 1}$

To determine

To graph: The function $f\left(x\right)=3{\left(x-2\right)}^2+1$, using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, $y=x^2$) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function.

Answer to Problem 52AYU

Domain of the function $f\left(x\right)=3{\left(x-2\right)}^2+1$ is $\left(-\infty ,\infty \right)$.

Range of the function $f\left(x\right)=3{\left(x-2\right)}^2+1$ is $\left[1,\infty \right)$.

Explanation of Solution

Given:

$f\left(x\right)=3{\left(x-2\right)}^2+1$

Graph:

Now use the following steps to obtain the graph of $f\left(x\right)$.

Step 1: The function $f\left(x\right)=x^2$ is the square function.

$f\left(x\right)=x^2$     square function

Step 2: To obtain the graph of $f\left(x\right)={\left(x-2\right)}^2$, replace $x$ by $x-2$ from each $y\text{-coordinate}$ on the graph of $f\left(x\right)=x^2$, that it is shifted right 2 units.

$f\left(x\right)={\left(x-2\right)}^2$     replace $x$ by $x-2$; Horizontal shift right 2 units.

Step 3: To obtain the graph of $f\left(x\right)=3{\left(x-2\right)}^2$, multiply each $y\text{-coordinate}$ of the graph of $f\left(x\right)={\left(x-2\right)}^2$, by 3, that it is vertically stretched by the factor of 3.

$f\left(x\right)=3{\left(x-2\right)}^2$     Multiply by 3, vertically stretched by a factor of 3

Step 4: To obtain the graph of $f\left(x\right)=3{\left(x-2\right)}^2+1$, add 1 from each $y\text{-coordinate}$ on the graph of $f\left(x\right)=3{\left(x-2\right)}^2$, that it is shifted up 1 unit.

$f\left(x\right)=3{\left(x-2\right)}^2+1$     add 1, vertically shift up 1 unit.

Interpretation:

Domain of the function $f\left(x\right)=3{\left(x-2\right)}^2+1$ is $\left(-\infty ,\infty \right)$.

Range of the function $f\left(x\right)=3{\left(x-2\right)}^2+1$ is $\left[1,\infty \right)$.