Question
What is the vertex form of the graphed equation? 4,8
(1 point)
Responses
y=2(x−4)^2−8
y equals 2 left parenthesis x minus 4 right parenthesis squared minus 8
y=2(x−4)^2+8
y equals 2 left parenthesis x minus 4 right parenthesis squared plus 8
y=−2(x−4)^2+8
y equals negative 2 left parenthesis x minus 4 right parenthesis squared plus 8
y=2(x+4)^2+8
y equals 2 left parenthesis x plus 4 right parenthesis squared plus 8
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(1 point)
Responses
y=2(x−4)^2−8
y equals 2 left parenthesis x minus 4 right parenthesis squared minus 8
y=2(x−4)^2+8
y equals 2 left parenthesis x minus 4 right parenthesis squared plus 8
y=−2(x−4)^2+8
y equals negative 2 left parenthesis x minus 4 right parenthesis squared plus 8
y=2(x+4)^2+8
y equals 2 left parenthesis x plus 4 right parenthesis squared plus 8
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Answers
GPT-4o mini
To find the vertex form of a quadratic equation given a vertex point, we can use the general form:
\[ y = a(x-h)^2 + k \]
where \((h, k)\) is the vertex of the parabola, and \(a\) determines the direction and width of the parabola.
In your case, the vertex is given as \((4, 8)\). So, \(h = 4\) and \(k = 8\). The equation becomes:
\[ y = a(x-4)^2 + 8 \]
Now we need to determine the value of \(a\). The choices provided suggest specific values for \(a\). Let's analyze them:
1. \(y = 2(x - 4)^2 - 8\) (vertex would be \((4, -8)\))
2. \(y = 2(x - 4)^2 + 8\) (vertex is \((4, 8)\))
3. \(y = -2(x - 4)^2 + 8\) (vertex is \((4, 8)\), opens down)
4. \(y = 2(x + 4)^2 + 8\) (vertex would be \((-4, 8)\))
Since we are looking for the vertex \( (4, 8) \) and it must have the \( +8 \) term to keep the vertex at the correct height in option 2 or 3, the form that matches our criteria considering the vertex position \( (4, 8) \) is:
\[ y = 2(x - 4)^2 + 8 \]
Thus, the correct response is:
**y = 2(x - 4)^2 + 8**
\[ y = a(x-h)^2 + k \]
where \((h, k)\) is the vertex of the parabola, and \(a\) determines the direction and width of the parabola.
In your case, the vertex is given as \((4, 8)\). So, \(h = 4\) and \(k = 8\). The equation becomes:
\[ y = a(x-4)^2 + 8 \]
Now we need to determine the value of \(a\). The choices provided suggest specific values for \(a\). Let's analyze them:
1. \(y = 2(x - 4)^2 - 8\) (vertex would be \((4, -8)\))
2. \(y = 2(x - 4)^2 + 8\) (vertex is \((4, 8)\))
3. \(y = -2(x - 4)^2 + 8\) (vertex is \((4, 8)\), opens down)
4. \(y = 2(x + 4)^2 + 8\) (vertex would be \((-4, 8)\))
Since we are looking for the vertex \( (4, 8) \) and it must have the \( +8 \) term to keep the vertex at the correct height in option 2 or 3, the form that matches our criteria considering the vertex position \( (4, 8) \) is:
\[ y = 2(x - 4)^2 + 8 \]
Thus, the correct response is:
**y = 2(x - 4)^2 + 8**