Suppose a parabola has vertex (5, –3) and also passes through the point (6, 1). Write the equation of the parabola in vertex form.
(1 point)
Responses
y = (x – 5)2 – 3
y = ( x – 5) 2 – 3
y = 4(x – 5)2 – 3
y = 4( x – 5) 2 – 3
y = 4(x + 5)2 – 3
y = 4( x + 5) 2 – 3
y = 4(x – 5)2 + 3
The correct equation of the parabola in vertex form is y = (x - 5)2 - 3.
To write the equation of a parabola in vertex form using the given vertex and a point on the parabola, follow these steps:
Step 1: Substitute the x-coordinate of the vertex, which is 5, into the equation for x.
Step 2: Substitute the y-coordinate of the vertex, which is -3, into the equation for y.
Step 3: Substitute the x-coordinate of the other given point, which is 6, into the equation for x.
Step 4: Substitute the y-coordinate of the other given point, which is 1, into the equation for y.
Step 5: Simplify the equation by performing any necessary operations.
Let's follow these steps:
Step 1: Substitute x = 5 into the equation:
y = (x - 5)^2 - 3
y = (5 - 5)^2 - 3
y = 0^2 - 3
y = -3
Step 2: Substitute y = -3 into the equation:
-3 = (x - 5)^2 - 3
-3 + 3 = (x - 5)^2
0 = (x - 5)^2
Step 3: Substitute x = 6 into the equation:
0 = (6 - 5)^2
0 = (1)^2
0 = 1
Step 4: Substitute y = 1 into the equation:
1 = (x - 5)^2 - 3
1 + 3 = (x - 5)^2
4 = (x - 5)^2
Step 5: Simplify the equation:
0 = (x - 5)^2
4 = (x - 5)^2
Therefore, the equation of the parabola in vertex form is y = (x - 5)^2.
Hence, the correct answer is:
y = (x - 5)^2 - 3.
To find the equation of the parabola in vertex form, we will use the vertex form equation:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
Given that the vertex is (5, -3), we can substitute these values into the equation:
y = a(x - 5)^2 - 3
Now, we need to find the value of 'a'. To do that, we can use the point (6, 1) that the parabola passes through. Substituting these values into the equation, we get:
1 = a(6 - 5)^2 - 3
Simplifying this equation, we have:
1 = a(1)^2 - 3
1 = a - 3
a = 4
Now that we know the value of 'a' is 4, we can substitute it back into the equation to get the final equation of the parabola:
y = 4(x - 5)^2 - 3
Therefore, the correct equation in vertex form is y = 4(x - 5)^2 - 3, which corresponds to the fourth option.