Question
Part 1 of 2 Determine whether the equation is linear or not. Then graph the equation by finding and plotting ordered-pair solutions. \( y=6 x^{2} \)
Ask by Parsons Norris. in the United States
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The equation \( y = 6x^{2} \) is **non-linear** and its graph is a parabola opening upward with its vertex at the origin. Here's a simplified graph:
```
y
|
24 | * *
|
|
18 |
|
|
12 |
|
|
6 | * *
|
|
0 |____*_____________________ x
-2 -1 0 1 2
```
Key points:
- Vertex at (0, 0)
- Symmetrical about the y-axis
- Points (-2, 24), (-1, 6), (0, 0), (1, 6), and (2, 24) lie on the graph.
Solution
### Determining Linearity and Graphing the Equation \( y = 6x^{2} \)
#### **1. Is the Equation Linear?**
**Definition of a Linear Equation:**
A linear equation in two variables, \( x \) and \( y \), has the general form:
\[
y = mx + b
\]
where:
- \( m \) is the slope,
- \( b \) is the y-intercept.
**Characteristics of Linear Equations:**
- The highest power of \( x \) and \( y \) is 1.
- The graph of a linear equation is a straight line.
**Analyzing \( y = 6x^{2} \):**
- The equation \( y = 6x^{2} \) has \( x \) raised to the power of 2.
- This means the equation is **quadratic**, not linear.
**Conclusion:**
- **Non-linear**.
#### **2. Graphing the Equation by Plotting Ordered Pairs**
Since the equation is quadratic, its graph will be a **parabola**. Here's how to graph it step-by-step:
**a. Create a Table of Values**
Choose various \( x \)-values and compute the corresponding \( y \)-values using the equation \( y = 6x^{2} \).
| \( x \) | \( y = 6x^{2} \) | Ordered Pair \( (x, y) \) |
|---|---|---|
| -2 | \( 6(-2)^{2} = 6 \times 4 = 24 \) | (-2, 24) |
| -1 | \( 6(-1)^{2} = 6 \times 1 = 6 \) | (-1, 6) |
| 0 | \( 6(0)^{2} = 0 \) | (0, 0) |
| 1 | \( 6(1)^{2} = 6 \times 1 = 6 \) | (1, 6) |
| 2 | \( 6(2)^{2} = 6 \times 4 = 24 \) | (2, 24) |
**b. Plot the Ordered Pairs**
Plot the points on a coordinate plane:
```
y
|
24 | * *
|
18 |
|
12 |
|
6 | * *
|
0 |________________________ x
-2 -1 0 1 2
```
**c. Draw the Parabola**
Connect the plotted points with a smooth, curved line forming a U-shaped parabola that opens upward.
**Key Features of the Graph:**
- **Vertex:** (0, 0) – the lowest point of the parabola.
- **Axis of Symmetry:** The vertical line \( x = 0 \).
- **Direction:** Opens **upward** because the coefficient of \( x^{2} \) is positive.
#### **Final Graph**
Here's a simplified representation of the graph:
```
y
|
24 | * *
|
|
18 |
|
|
12 |
|
|
6 | * *
|
|
0 |____*_____________________ x
-2 -1 0 1 2
```
- The points \((-2, 24)\) and \((2, 24)\) are symmetrical about the y-axis.
- The points \((-1, 6)\) and \((1, 6)\) are also symmetrical.
- The origin \((0, 0)\) is the vertex of the parabola.
Feel free to use graphing software or graph paper for a more accurate and detailed graph!
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
The equation \( y = 6x^2 \) is not linear because its variable \( x \) is raised to the second power, indicating that it represents a quadratic function. Linear equations take the form \( y = mx + b \), where the highest exponent of \( x \) is 1. To graph \( y = 6x^2 \), you can find ordered pairs by substituting various values of \( x \). For example, with \( x = -1 \), \( y = 6(-1)^2 = 6 \); for \( x = 0 \), \( y = 6(0)^2 = 0 \); and with \( x = 1 \), \( y = 6(1)^2 = 6 \). Plotting the points \((-1, 6)\), \((0, 0)\), and \((1, 6)\) will reveal the parabolic shape of the graph!
