Graph Using a Table of Values Y=2x^2+1

Solution:

Step 1:

Insert $x=-2$ into the equation to compute $y$. $Y=2(-2{)}^2+1$

Step 2:

Perform the simplification of $2(-2{)}^2+1$.

Step 2.1:

Break down the expression term by term.

Step 2.1.1:

Square the number $-2$. $Y=2\cdot 4+1$

Step 2.1.2:

Multiply $2$ by the result of squaring $-2$. $Y=8+1$

Step 2.2:

Combine the sum of $8$ and $1$. $Y=9$

Step 3:

Replace $x$ with $-1$ to determine $y$. $Y=2(-1{)}^2+1$

Step 4:

Simplify the expression $2(-1{)}^2+1$.

Step 4.1:

Simplify the components of the expression.

Step 4.1.1:

Square the number $-1$. $Y=2\cdot 1+1$

Step 4.1.2:

Multiply $2$ by the squared result. $Y=2+1$

Step 4.2:

Sum the numbers $2$ and $1$. $Y=3$

Step 5:

Use $x=0$ in the equation to find $y$. $Y=2(0{)}^2+1$

Step 6:

Simplify the term $2(0{)}^2+1$.

Step 6.1:

Simplify each part of the term.

Step 6.1.1:

Zero raised to any positive exponent is $0$. $Y=2\cdot 0+1$

Step 6.1.2:

Multiply $2$ by $0$. $Y=0+1$

Step 6.2:

Add the numbers $0$ and $1$. $Y=1$

Step 7:

Put $x=1$ into the equation to solve for $y$. $Y=2(1{)}^2+1$

Step 8:

Simplify $2(1{)}^2+1$.

Step 8.1:

Simplify each component of the expression.

Step 8.1.1:

One raised to any power is still one. $Y=2\cdot 1+1$

Step 8.1.2:

Multiply $2$ by $1$. $Y=2+1$

Step 8.2:

Sum up $2$ and $1$. $Y=3$

Step 9:

Substitute $x=2$ into the equation to find $y$. $Y=2(2{)}^2+1$

Step 10:

Simplify the expression $2(2{)}^2+1$.

Step 10.1:

Break down the expression term by term.

Step 10.1.1:

Apply exponent rules to multiply $2$ by $(2{)}^2$.

Step 10.1.1.1:

Multiply $2$ by $(2{)}^2$.

Step 10.1.1.1.1:

Raise $2$ to the first power. $Y=2^1(2{)}^2+1$

Step 10.1.1.1.2:

Apply the power rule $a^m{a}^n=a^{m+n}$ to combine the exponents. $Y=2^{1+2}+1$

Step 10.1.1.2:

Add the exponents $1$ and $2$. $Y=2^3+1$

Step 10.1.2:

Raise $2$ to the third power. $Y=8+1$

Step 10.2:

Add the numbers $8$ and $1$. $Y=9$

Step 11:

Create a table with the calculated values for graphing the equation.

$$\begin{array}{cc}x& y\\ -2& 9\\ -1& 3\\ 0& 1\\ 1& 3\\ 2& 9\end{array}$$

Step 12:

Plot the points from the table on a graph to visualize the quadratic function.

Knowledge Notes:

The problem involves graphing a quadratic function, which is a polynomial function of degree two. The standard form of a quadratic function is $y=a{x}^2+bx+c$, where $a$, $b$, and $c$ are constants, and $a\ne 0$. The graph of a quadratic function is a parabola.

To graph a quadratic function using a table of values:

1. Choose a set of $x$ values.

2. Substitute each $x$ value into the quadratic equation to find the corresponding $y$ value.

3. Simplify the equation at each step to obtain the $y$ values.

4. Plot the $(x,y)$ points on a coordinate plane.

5. Connect the points to form a parabola.

In this case, the quadratic function is $y=2x^2+1$, and the process involves calculating $y$ for a set of $x$ values, which are $-2$, $-1$, $0$, $1$, and $2$. The simplification steps include squaring the $x$ value, multiplying by the coefficient $2$, and adding the constant $1$. The power rule for exponents, $a^m{a}^n=a^{m+n}$, is used when simplifying expressions involving exponents. The resulting points are then used to plot the graph of the quadratic function.