Question
Which of the relationships below represents a function with a greater rate of change than the function y, equals, minus, 4, x, plus, 4y=−4x+4? A x y B xx yy 66 minus, 33−33 88 minus, 43−43 1010 minus, 53−53 1212 minus, 63−63 C xx yy minus, 2−2 55 00 minus, 3−3 22 minus, 11−11 44 minus, 19−19 D x y Answer
Answers
GPT-4o mini
To determine which relationship represents a function with a greater rate of change than the function given by the equation \(-4x + 4y = -4x + 4\), we should first rearrange this equation into slope-intercept form (y = mx + b), where m is the slope (rate of change).
Starting with \(-4x + 4y = -4x + 4\):
1. Add \(4x\) to both sides:
\[
4y = 4
\]
2. Divide both sides by 4:
\[
y = 1
\]
The equation simplifies to \(y = 1\), which has a slope of \(0\) (meaning it is a horizontal line).
Now, we are looking for relationships (or functions) that have a greater rate of change (slope) than \(0\). Any positive slope will be greater than this horizontal line.
### Let's evaluate each option for their slopes:
**A.**
- \((x, y)\): \((6, -3)\), \((8, -4)\), \((10, -5)\), \((12, -6)\)
- The change in \(y\) is \(-1\) for each \(x\) increment of \(2\).
- Slope \(m = \frac{\Delta y}{\Delta x} = \frac{-1}{2} = -0.5\) (not greater than 0)
**B.**
- \((x, y)\): \((-2, -2)\), \((5, 0)\), \((-3, -3)\), \((2, -1)\), \((-1, -1)\)
- The slope calculation is not consistent; we need one that is representative. For example:
- Between \((-3, -3)\) and \((2, -1)\), the change is \(2\) in \(y\) and \(5\) in \(x\) gives \(\frac{-1 - (-3)}{2 - (-3)} = \frac{2}{5} = 0.4\) (greater than 0)
**C.**
- \((x, y)\): \((-2, -2)\), \((5, 0)\), \((-3, -3)\), \((2, -1)\), \((-1, -1)\)
- Again, as we calculated, these accounted multiple points but should be looked further in a slope way. Averageing shows could likely lie below zero, as example pairs show negative slopes.
**D.**
- Without specific points or pairs to assess here means no numerical representation shown.
### Conclusion:
From the evaluations of the pairs in options A, B, and C, it becomes apparent option B had at least one calculation for slope greater than 0, making it our answer. A check for all pairs shows a slip might steer for positive, hence
**Answer: B** has sections potentially yielding values above ground zero, without clear instances pointing back, we tentatively mark that option until revaluation proposes with explicit ranges from options laid out.
If there's precise relevance missing in others not assuming standard placement we'd gain complete result in raw pair entries.
Starting with \(-4x + 4y = -4x + 4\):
1. Add \(4x\) to both sides:
\[
4y = 4
\]
2. Divide both sides by 4:
\[
y = 1
\]
The equation simplifies to \(y = 1\), which has a slope of \(0\) (meaning it is a horizontal line).
Now, we are looking for relationships (or functions) that have a greater rate of change (slope) than \(0\). Any positive slope will be greater than this horizontal line.
### Let's evaluate each option for their slopes:
**A.**
- \((x, y)\): \((6, -3)\), \((8, -4)\), \((10, -5)\), \((12, -6)\)
- The change in \(y\) is \(-1\) for each \(x\) increment of \(2\).
- Slope \(m = \frac{\Delta y}{\Delta x} = \frac{-1}{2} = -0.5\) (not greater than 0)
**B.**
- \((x, y)\): \((-2, -2)\), \((5, 0)\), \((-3, -3)\), \((2, -1)\), \((-1, -1)\)
- The slope calculation is not consistent; we need one that is representative. For example:
- Between \((-3, -3)\) and \((2, -1)\), the change is \(2\) in \(y\) and \(5\) in \(x\) gives \(\frac{-1 - (-3)}{2 - (-3)} = \frac{2}{5} = 0.4\) (greater than 0)
**C.**
- \((x, y)\): \((-2, -2)\), \((5, 0)\), \((-3, -3)\), \((2, -1)\), \((-1, -1)\)
- Again, as we calculated, these accounted multiple points but should be looked further in a slope way. Averageing shows could likely lie below zero, as example pairs show negative slopes.
**D.**
- Without specific points or pairs to assess here means no numerical representation shown.
### Conclusion:
From the evaluations of the pairs in options A, B, and C, it becomes apparent option B had at least one calculation for slope greater than 0, making it our answer. A check for all pairs shows a slip might steer for positive, hence
**Answer: B** has sections potentially yielding values above ground zero, without clear instances pointing back, we tentatively mark that option until revaluation proposes with explicit ranges from options laid out.
If there's precise relevance missing in others not assuming standard placement we'd gain complete result in raw pair entries.