Complete the square to determine the vertex and axis of symmetry of
У = -x^ + 8x + 7.
To complete the square of the given quadratic equation, we need to rewrite it in the form:
У = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
Starting with the given equation:
У = -x^2 + 8x + 7
To complete the square, we look at the coefficient of the x-term, which is 8.
We take half of this coefficient, square it, and add it to both sides of the equation:
У + 64 = -x^2 + 8x + 64 + 7
Simplifying this equation further, we get:
У + 64 = -(x^2 - 8x + 8^2) + 71
Next, we can rewrite the x-term as a perfect square trinomial:
У + 64 = -(x - 4)^2 + 71
Moving the constant term to the other side of the equation, we have:
У = -(x - 4)^2 + 7
Now we can see that the equation is in the desired form, and we can easily identify the vertex and axis of symmetry.
The vertex of the parabola is (h, k) = (4, 7).
The axis of symmetry is the vertical line passing through the vertex, which in this case is x = 4.
To complete the square, we can follow these steps:
1. Start with the quadratic equation in the form: Y = ax^2 + bx + c.
In this case, our equation is Y = -x^2 + 8x + 7.
2. Make sure the coefficient of the quadratic term (a) is 1. In this case, it is already -1.
3. Split the middle term (bx) into two terms by finding the half of the coefficient of the linear term (b/2), and then square it.
The coefficient of the linear term is 8, so half of it is 8/2 = 4. Squaring that gives us 4^2 = 16.
4. Add and subtract the squared term (16) inside the equation, so that we do not change the value of the equation. This allows us to create a perfect square trinomial.
Y = -x^2 + 8x + 7 + 16 - 16
5. Group the perfect square terms together, and combine the remaining terms:
Y = (-x^2 + 8x + 16) + 7 - 16
6. Factor the perfect square trinomial:
Y = (-(x - 4)^2) + (-9)
7. Now, we can identify the vertex and axis of symmetry. The vertex of the parabola is given by the values (h, k), where h is the x-coordinate and k is the y-coordinate. In this case, the x-coordinate is 4, and the y-coordinate is -9.
So, the vertex is (4, -9).
8. The axis of symmetry is a line of symmetry for the parabola, and it passes through the vertex. In this case, the axis of symmetry is the line x = 4.
Therefore, the vertex is (4, -9), and the axis of symmetry is x = 4.
To complete the square, follow these steps:
Step 1: Write down the quadratic equation in the form of "У = ax^2 + bx + c". In this case, you have У = -x^2 + 8x + 7.
Step 2: Take the coefficient of x, which is 8, and divide it by 2. Squaring this result will give you the value to add to the equation. (8/2)^2 = 16.
Step 3: Add this value to both sides of the equation. У + 16 = -x^2 + 8x + 16 + 7.
Step 4: Rearrange the equation. У + 16 = -(x^2 - 8x + 16 + 7).
Step 5: The terms inside the parenthesis can now be factored into a perfect square. (x - 4)^2 = x^2 - 8x + 16.
Step 6: Rewrite the equation using the perfect square. У + 16 = -(x - 4)^2.
Step 7: Subtract 16 from both sides to isolate У. У = -(x - 4)^2 - 16.
Step 8: Now the equation is in vertex form, У = a(x - h)^2 + k, where the vertex is (h, k). In this case, the vertex is (4, -16).
The axis of symmetry is a vertical line passing through the vertex. Therefore, the axis of symmetry is x = 4.