Question

Which points are on the plane curve described by the following set of parametric equations? Select all that apply.

x = 3t + 4 and y equals 2 t squared
(1 point)
Responses

left parenthesis 1 comma negative 2 right parenthesis
Image with alt text: left parenthesis 1 comma negative 2 right parenthesis

left parenthesis 1 comma 2 right parenthesis
Image with alt text: left parenthesis 1 comma 2 right parenthesis

left parenthesis 1 comma 7 right parenthesis
Image with alt text: left parenthesis 1 comma 7 right parenthesis

left parenthesis 2 comma 10 right parenthesis
Image with alt text: left parenthesis 2 comma 10 right parenthesis

left parenthesis 7 comma 2 right parenthesis

Answer 1

To determine which points are on the plane curve described by the parametric equations \( x = 3t + 4 \) and \( y = 2t^2 \), we can perform the following steps:

1. **Solve for \( t \)** using the \( x \) equation.
2. **Substitute \( t \)** into the \( y \) equation.
3. **Check each point against the resulting values** of \( x \) and \( y \).

Let's proceed step-by-step:

### Step 1: Solve for \( t \)
From the equation \( x = 3t + 4 \), we can solve for \( t \):
\[
t = \frac{x - 4}{3}
\]

### Step 2: Substitute \( t \) into the \( y \) equation
The equation for \( y \) is \( y = 2t^2 \). Substitute \( t = \frac{x - 4}{3} \) into the \( y \) equation:
\[
y = 2\left(\frac{x - 4}{3}\right)^2
\]
\[
y = 2\left(\frac{(x - 4)^2}{9}\right) = \frac{2(x - 4)^2}{9}
\]

### Step 3: Check Each Point
Now we will substitute the \( x \) values of the given points into the equation for \( y \) and check if the \( y \) value matches the given points:

1. **Point \( (1, -2) \)**:
\[
x = 1 \Rightarrow y = \frac{2(1 - 4)^2}{9} = \frac{2(9)}{9} = 2
\]
Not a match (2 vs -2).

2. **Point \( (1, 2) \)**:
\[
x = 1 \Rightarrow y = \frac{2(1 - 4)^2}{9} = \frac{2(9)}{9} = 2
\]
A match (2 vs 2).

3. **Point \( (1, 7) \)**:
\[
x = 1 \Rightarrow y = \frac{2(1 - 4)^2}{9} = \frac{2(9)}{9} = 2
\]
Not a match (2 vs 7).

4. **Point \( (2, 10) \)**:
\[
x = 2 \Rightarrow y = \frac{2(2 - 4)^2}{9} = \frac{2(4)}{9} = \frac{8}{9}
\]
Not a match (approximately 0.89 vs 10).

5. **Point \( (7, 2) \)**:
\[
x = 7 \Rightarrow y = \frac{2(7 - 4)^2}{9} = \frac{2(3^2)}{9} = \frac{18}{9} = 2
\]
A match (2 vs 2).

### Final Matches
The points on the curve are:
- **\( (1, 2) \)**
- **\( (7, 2) \)**

Thus, the responses that fit the set of parametric equations are:
- \( (1, 2) \)
- \( (7, 2) \)