Hey I am working on math homework that involves "Standard Form to Vertex Form by Averaging." My class is just starting to learn this today, but I am totally lost still on what to do. Can you please help? The formula for this is y=a(x - h)^2 + k
the h is the x-value vertex and the k is the y-value vertex.
I got the first one done and it looks like this:
y=-x^2 + 2x +3
-1(x^2 - 2x - 3)
-1(x + 1)(x - 3)
x= -1 and 3
y=-(-1)^2 + 2(-1) + 3
y=-1 + 2 + 3
y= 4
then
h=-1 and k=4
I have many other like this but this was the only one i understood. If any one knows this process i'd be so thankful for some help! Thank you!
Sure! I'd be happy to help you understand the process of converting a quadratic equation from standard form to vertex form by averaging.
To convert the equation from standard form (y = ax^2 + bx + c) to vertex form (y = a(x - h)^2 + k), you follow these steps:
Step 1: Identify the values of a, b, and c in the given equation.
In your equation, y = -x^2 + 2x + 3, a = -1, b = 2, and c = 3.
Step 2: Find the x-coordinate of the vertex, denoted as h.
The x-coordinate of the vertex (h) can be found using the formula h = -b / (2a).
Substituting the values from your equation, h = -(2) / (2*(-1)) = -1.
Step 3: Substitute the value of h into the equation and simplify.
Substituting h = -1 into the equation, we get:
y = -x^2 + 2x + 3 becomes y = -x^2 + 2x + 3 = -1(-1)^2 + 2(-1) + 3 = -1 + (-2) + 3 = 0.
Step 4: Find the y-coordinate of the vertex, denoted as k.
Now that you have the x-coordinate of the vertex (h = -1), substitute it back into the simplified equation to find the y-coordinate (k).
Using the value h = -1, we get k = 0.
Step 5: Write the equation in vertex form.
Substituting the values of h and k into the vertex form equation y = a(x - h)^2 + k, we get:
y = -1(x - (-1))^2 + 0 which simplifies to y = -(x + 1)^2.
So, the vertex form of the equation y = -x^2 + 2x + 3 is y = -(x + 1)^2.
I hope this helps! Let me know if you have any further questions.
Of course, I can help you with converting a quadratic equation from standard form to vertex form.
In the standard form of a quadratic equation, y = ax^2 + bx + c, the vertex form is given by y = a(x - h)^2 + k. In vertex form, (h, k) represents the coordinates of the vertex.
To convert a quadratic equation from standard form to vertex form by averaging, follow these steps:
Step 1: Rewrite the equation in a factored form (if possible) by factoring out the coefficient of x^2. This step involves factoring out any common factors, if applicable.
In your example, the equation is:
y = -x^2 + 2x + 3
Step 2: Factor out the coefficient of x^2 from the equation. In this case, the coefficient is -1:
y = -1(x^2 - 2x - 3)
Step 3: Factor the quadratic expression inside the parentheses. Look for two numbers that multiply to give the constant term (-3) and add up to give the coefficient of x (-2). In this case, the factors are -3 and +1:
y = -1(x + 1)(x - 3)
Step 4: Find the x-intercepts, which are values of x that make y equal to zero. Set each factor equal to zero and solve for x:
x + 1 = 0 ==> x = -1
x - 3 = 0 ==> x = 3
Step 5: Average the x-intercepts to find the x-value of the vertex. Add the two x-intercepts and divide by 2:
h = (-1 + 3) / 2 = 2 / 2 = 1
Step 6: Substitute the x-value of the vertex back into the factored form to find the y-value of the vertex. Replace x with the value of h (in this case, 1), and solve for y:
y = -(1 + 1)(1 - 3)
y = -2*(-2)
y = 4
Therefore, the vertex of the quadratic equation y = -x^2 + 2x + 3, when converted to vertex form, is (h, k) = (1, 4).
You can follow the same steps to convert other quadratic equations from standard form to vertex form. Hope this helps you understand the process better.