Find the coordinates of the vertex and the equation of the axis of symmetry of the parabola given by the equation.
x = −1/3y^2+4
vertex
(x, y) =
axis of symmetry
Sketch the graph.
recall that the vertex of
x-h = a(y-k)^2
is at (h,k). Look at your function and you see that
x-4 = -1/3 (y-0)^2
so the vertex is at (4,0)
Now use your knowledge that the vertex lies on the axis of symmetry, and you're almost home free.
To find the coordinates of the vertex and the equation of the axis of symmetry of the parabola given by the equation x = -1/3y^2 + 4, we need to manipulate the equation into the standard form of a parabola, which is y = ax^2 + bx + c.
Step 1: Rearrange the given equation:
x = -1/3y^2 + 4
Multiply both sides by -3:
-3x = y^2 - 12
Rearrange to make y^2 the subject:
y^2 = -3x + 12
Step 2: We can now rewrite the equation in the standard form by completing the square. To do this, we need to add and subtract a suitable constant on the right side of the equation. In this case, the constant we need to add and subtract is (12/2)^2 = 36:
y^2 = -3x + 12 + 36 - 36
Simplify the right side:
y^2 = -3x + 48 - 36
y^2 = -3x + 12
Step 3: Rewrite the equation in the standard form:
y^2 = -3x + 12
y^2 = -3(x - 4)
Now, we can see that the equation is in the form y = ax^2 + bx + c, where a = -3, b = 0, and c = 12.
The coordinates of the vertex can be obtained by using the formula x = -b/2a and plugging in the values of a and b from the equation.
x = -0/(2*(-3))
x = 0/(-6) = 0
To find the corresponding y-coordinate, we substitute the value of x = 0 back into the equation y^2 = -3(x - 4):
y^2 = -3(0 - 4)
y^2 = -3(-4)
y^2 = 12
Taking the square root of both sides, we get:
y = ±√12
y ≈ ±3.464 (rounded to three decimal places)
Therefore, the vertex of the parabola is (0, ±3.464).
The equation of the axis of symmetry is simply the x-coordinate of the vertex. So, in this case, the equation of the axis of symmetry is x = 0.
To sketch the graph, plot the vertex (0, ±3.464) and draw the axis of symmetry, which is the vertical line x = 0. Then, choose a few more points on either side of the vertex and plot them. Finally, connect the plotted points with a smooth curve, representing the shape of the parabola.