Graph Using a Table of Values Y=2x^2+1
Explanation of the question:
The question is asking for a method to create a visual representation, specifically a graph, of the quadratic function Y=2x^2+1 using a set of selected values for 'x'. To do this, you need to create a table where you choose different 'x' values, substitute them into the equation to calculate the corresponding 'y' values, and then plot these (x, y) pairs on a coordinate plane. By connecting the points, you would illustrate the parabolic shape defined by the quadratic function.
$Y = 2 x^{2} + 1$
Answer
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.
$$ \begin{array}{c|c} x & y \\ \hline -2 & 9 \\ -1 & 3 \\ 0 & 1 \\ 1 & 3 \\ 2 & 9 \\ \end{array} $$
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 $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a \neq 0$. The graph of a quadratic function is a parabola.
To graph a quadratic function using a table of values:
Choose a set of $x$ values.
Substitute each $x$ value into the quadratic equation to find the corresponding $y$ value.
Simplify the equation at each step to obtain the $y$ values.
Plot the $(x, y)$ points on a coordinate plane.
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.
