Suppose a parabola has an axis of symmetry at x equals negative eight , a maximum height of 2, and passes through the point Left parenthesis negative 7 comma negative 1 right parenthesis. Write the equation of the parabola in vertex form.
(1 point)
Responses
y equals negative 7 left parenthesis x plus 2 right parenthesis squared minus 8
Image with alt text: y equals negative 7 left parenthesis x plus 2 right parenthesis squared minus 8
y equals negative 0.01 left-parenthesis x minus 8 right-parenthesis squared plus 2
Image with alt text: y equals negative 0.01 left-parenthesis x minus 8 right-parenthesis squared plus 2
y equals negative 3 left parenthesis x plus 8 right parenthesis squared plus 2
Image with alt text: y equals negative 3 left parenthesis x plus 8 right parenthesis squared plus 2
y equals negative 1.5 left parenthesis x minus 8 right parenthesis squared plus 2
The correct response is:
y equals negative 3 left parenthesis x plus 8 right parenthesis squared plus 2
To write the equation of the parabola in vertex form, we can use the formula:
\(y = a(x - h)^2 + k\)
where (h, k) represents the coordinates of the vertex.
Given that the axis of symmetry is at x = -8, we know that the x-coordinate of the vertex is -8. Also, the maximum height of the parabola is 2, so the y-coordinate of the vertex is 2.
Therefore, the equation of the parabola in vertex form is:
\(y = a(x - (-8))^2 + 2\)
which simplifies to:
\(y = a(x + 8)^2 + 2\)
Now we need to find the value of 'a'. We can use the fact that the parabola passes through the point (-7, -1). Substituting these values into the equation, we have:
\(-1 = a((-7) + 8)^2 + 2\)
Simplifying further:
\(-1 = a(1)^2 + 2\)
\(-1 = a + 2\)
\(a = -3\)
Therefore, the equation of the parabola in vertex form is:
\(y = -3(x + 8)^2 + 2\)
So the correct option is:
y equals negative 3 left parenthesis x plus 8 right parenthesis squared plus 2
To write the equation of the parabola in vertex form, we need to know the coordinates of the vertex. In this case, the given information tells us that the axis of symmetry is at x = -8, and the vertex lies on the axis of symmetry. Since the parabola has a maximum height of 2, the vertex is at (-8, 2).
The vertex form of a parabola's equation is given by:
y = a(x - h)^2 + k
Where (h, k) represents the vertex of the parabola.
Plugging in the values for the vertex, we get:
y = a(x - (-8))^2 + 2
Simplifying,
y = a(x + 8)^2 + 2
Now, we need to find the value of 'a'. To do this, we can use the point (-7, -1) which lies on the parabola. Substituting these values into the equation, we get:
-1 = a(-7 + 8)^2 + 2
Simplifying further,
-1 = a(1)^2 + 2
-1 = a + 2
a = -3
Now, substituting the value of 'a' (-3) back into the equation, we get the final equation of the parabola:
y = -3(x + 8)^2 + 2
Therefore, the correct answer is: y = -3(x + 8)^2 + 2.