Question
Which equation is equivalent to the formula below?
y = a(x - h)^2 + k
A. x = ± √y - k/a - h
B. a = y - k/(x - h)^2
C. h = x - (y - k/a)^2
D. k = y + (x - h)^2
y = a(x - h)^2 + k
A. x = ± √y - k/a - h
B. a = y - k/(x - h)^2
C. h = x - (y - k/a)^2
D. k = y + (x - h)^2
Answers
GPT-4o mini
To find the equation equivalent to the formula \( y = a(x - h)^2 + k \), we can manipulate this equation to isolate different variables. Let's rearrange the equation step by step.
Starting with the original equation:
\[ y = a(x - h)^2 + k \]
1. Subtract \( k \) from both sides:
\[ y - k = a(x - h)^2 \]
2. Divide both sides by \( a \):
\[ \frac{y - k}{a} = (x - h)^2 \]
3. Take the square root of both sides. Remember to consider both the positive and negative roots because we're dealing with a square:
\[ \sqrt{\frac{y - k}{a}} = x - h \quad \text{or} \quad -\sqrt{\frac{y - k}{a}} = x - h \]
This can be simplified to:
\[ x = h \pm \sqrt{\frac{y - k}{a}} \]
Now, let's rewrite this in a form that matches one of the answer choices. To isolate \( x \):
\[ x = h + \sqrt{\frac{y - k}{a}} \quad \text{or} \quad x = h - \sqrt{\frac{y - k}{a}} \]
Therefore, we have:
\[ x = h \pm \sqrt{\frac{y - k}{a}} \]
However, we have a choice in the question that suggests a possible manipulation of this. Comparing with the provided options, let's rearrange the step before taking the square root.
From:
\[ \frac{y - k}{a} = (x - h)^2 \]
If we take \( (x - h)^2 = \frac{y - k}{a} \) and move \( y \) and \( k \) terms:
This leads us to manipulate the equation further. Dividing \( \frac{y - k}{a} \) can give us possible terms like:
Let’s check if any options match:
A. **x = ± √(y - k/a) - h**
B. **a = (y - k)/(x - h)^2**
C. **h = x - (y - k/a)^2**
D. **k = y + (x - h)^2**
Indeed choice **A matches perfectly**, upon rearrangement and isolation of \( x \).
Thus, the correct choice is:
**A. x = ± √(y - k/a) - h**
Starting with the original equation:
\[ y = a(x - h)^2 + k \]
1. Subtract \( k \) from both sides:
\[ y - k = a(x - h)^2 \]
2. Divide both sides by \( a \):
\[ \frac{y - k}{a} = (x - h)^2 \]
3. Take the square root of both sides. Remember to consider both the positive and negative roots because we're dealing with a square:
\[ \sqrt{\frac{y - k}{a}} = x - h \quad \text{or} \quad -\sqrt{\frac{y - k}{a}} = x - h \]
This can be simplified to:
\[ x = h \pm \sqrt{\frac{y - k}{a}} \]
Now, let's rewrite this in a form that matches one of the answer choices. To isolate \( x \):
\[ x = h + \sqrt{\frac{y - k}{a}} \quad \text{or} \quad x = h - \sqrt{\frac{y - k}{a}} \]
Therefore, we have:
\[ x = h \pm \sqrt{\frac{y - k}{a}} \]
However, we have a choice in the question that suggests a possible manipulation of this. Comparing with the provided options, let's rearrange the step before taking the square root.
From:
\[ \frac{y - k}{a} = (x - h)^2 \]
If we take \( (x - h)^2 = \frac{y - k}{a} \) and move \( y \) and \( k \) terms:
This leads us to manipulate the equation further. Dividing \( \frac{y - k}{a} \) can give us possible terms like:
Let’s check if any options match:
A. **x = ± √(y - k/a) - h**
B. **a = (y - k)/(x - h)^2**
C. **h = x - (y - k/a)^2**
D. **k = y + (x - h)^2**
Indeed choice **A matches perfectly**, upon rearrangement and isolation of \( x \).
Thus, the correct choice is:
**A. x = ± √(y - k/a) - h**