12. Explain how the factored form of a quadratic equation, Y=a ( X−S )( X− T), can be used to calculate the value of the vertex. Be sure to describe the mathematical steps involved in this process
The factored form of a quadratic equation is helpful in determining the vertex of the parabola because it allows us to easily identify the x-coordinates of the two points where the parabola intersects the x-axis. These two x-intercepts, S and T, are equidistant from the vertex.
To calculate the x-coordinate of the vertex, we simply need to find the midpoint of S and T. This can be done using the formula x = (S + T)/2.
Once we have the x-coordinate of the vertex, we can substitute it into the original equation to find the y-coordinate. This will give us the coordinates of the vertex (x, y).
For example, if we have the factored form Y = 2 (X−3)(X−7), we know that the x-intercepts are at X=3 and X=7. Therefore, the x-coordinate of the vertex is x = (3 + 7)/2 = 5.
To find the y-coordinate of the vertex, we can substitute x = 5 into the original equation:
Y = 2 (5−3)(5−7)
Y = 2 (2)(-2)
Y = -8
Therefore, the vertex of the parabola is located at (5, -8).
To find the value of the vertex using the factored form of a quadratic equation, Y = a(X - S)(X - T), you need to determine the coordinate point (h, k), where h represents the X-coordinate of the vertex and k represents the Y-coordinate.
Follow these steps to calculate the value of the vertex:
Step 1: Identify the constant values a, S, and T from the given quadratic equation in factored form.
Step 2: To calculate the X-coordinate of the vertex (h), use the formula:
h = (S + T) / 2
Add the values of S and T together, and then divide by 2 to obtain the X-coordinate of the vertex.
Step 3: Substitute the calculated value of h back into the original quadratic equation to find the Y-coordinate (k).
Replace X with the X-coordinate (h) in the factored form equation to obtain:
Y = a(h - S)(h - T)
Simplify the equation by performing the subtraction in the brackets and multiplying. This will give you the equation in the form:
Y = ah² - a(hS + hT) + aST
Step 4: Simplify the equation further if possible and express it in standard form.
To find the simplified equation, distribute the constant 'a' across the terms in the equation:
Y = ah² - ahS - ahT + aST
Step 5: Identify the Y-coordinate (k) by looking at the constant term in the simplified equation.
The Y-coordinate (k) of the vertex is equal to the constant term in the equation. Hence, k = aST.
So, the value of the vertex is (h, k), where h = (S + T) / 2 and k = aST.
By following these steps, you can calculate the value of the vertex using the factored form of a quadratic equation.