Question
Given the table of values below, which equation best represents the information?
x y
150 300
250 500
360 720
480 960
f( x) = 2 x - 100
f( x) = 2 x
f( x) = x + 150
f( x) = x/2
x y
150 300
250 500
360 720
480 960
f( x) = 2 x - 100
f( x) = 2 x
f( x) = x + 150
f( x) = x/2
Answers
Answered by
GPT-4o mini
To find the equation that best represents the relationship between \( x \) and \( y \) in the table, we can analyze the provided values:
1. **Identifying the relationship:** Let's check how \( y \) changes with respect to \( x \):
- For \( x = 150 \), \( y = 300 \)
- For \( x = 250 \), \( y = 500 \)
- For \( x = 360 \), \( y = 720 \)
- For \( x = 480 \), \( y = 960 \)
Observing the changes in \( x \) and \( y \):
- The relationship appears to be linear, as \( y \) seems to double when \( x \) is increased in steps.
2. **Calculating the ratio:**
\[
\frac{y}{x} = \frac{300}{150} = 2, \quad \frac{500}{250} = 2, \quad \frac{720}{360} = 2, \quad \frac{960}{480} = 2
\]
The ratio \( \frac{y}{x} = 2 \) holds true for all data points.
3. **Finding the equation:**
Since \( \frac{y}{x} = 2 \), it can be rewritten as:
\[
y = 2x
\]
4. **Testing the options:**
- \( f(x) = 2x - 100 \): For \( x = 150 \), \( y = 2(150) - 100 = 300 - 100 = 200 \) (not correct).
- \( f(x) = 2x \): For \( x = 150 \), \( y = 2(150) = 300 \) (correct).
- \( f(x) = x + 150 \): For \( x = 150 \), \( y = 150 + 150 = 300 \) (correct but does not hold for all points).
- \( f(x) = x/2 \): For \( x = 150 \), \( y = 150/2 = 75 \) (not correct).
Thus, the best equation that represents the given table of values is:
\[
\boxed{f(x) = 2x}
\]
1. **Identifying the relationship:** Let's check how \( y \) changes with respect to \( x \):
- For \( x = 150 \), \( y = 300 \)
- For \( x = 250 \), \( y = 500 \)
- For \( x = 360 \), \( y = 720 \)
- For \( x = 480 \), \( y = 960 \)
Observing the changes in \( x \) and \( y \):
- The relationship appears to be linear, as \( y \) seems to double when \( x \) is increased in steps.
2. **Calculating the ratio:**
\[
\frac{y}{x} = \frac{300}{150} = 2, \quad \frac{500}{250} = 2, \quad \frac{720}{360} = 2, \quad \frac{960}{480} = 2
\]
The ratio \( \frac{y}{x} = 2 \) holds true for all data points.
3. **Finding the equation:**
Since \( \frac{y}{x} = 2 \), it can be rewritten as:
\[
y = 2x
\]
4. **Testing the options:**
- \( f(x) = 2x - 100 \): For \( x = 150 \), \( y = 2(150) - 100 = 300 - 100 = 200 \) (not correct).
- \( f(x) = 2x \): For \( x = 150 \), \( y = 2(150) = 300 \) (correct).
- \( f(x) = x + 150 \): For \( x = 150 \), \( y = 150 + 150 = 300 \) (correct but does not hold for all points).
- \( f(x) = x/2 \): For \( x = 150 \), \( y = 150/2 = 75 \) (not correct).
Thus, the best equation that represents the given table of values is:
\[
\boxed{f(x) = 2x}
\]
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