Answer:
Vertex form: 5(x + 1)^2 + 2; Vertex: (-1, 2)
Step-by-step explanation:
To convert the quadratic function to vertex form you need to first isolate the variable x^2. To do this factor out 5 from only the first two terms in the equation.
y = 5(x^2 + 2x) + 7
Now use (b/2)^2 to complete the square, where b = 2.
Add 1 inside the parentheses and subtract 1 (times 5 since the 1 being added is also being multiplied by 5) outside the parentheses.
This equation:
5(x^2 + 2x) + 7
becomes:
5(x^2 + 2x + 1) + 7 - 1(5)
Factor the trinomial inside the parentheses by seeing what two of the same numbers add to b (2) and multiply to 1 (c). This number is a positive 1.
5(x + 1)^2 + 7 - 1(5)
Now simplify the entire equation.
5(x + 1)^2 + 2 is the vertex form of this quadratic function.
Vertex form is a(x - h)^2 + k, where (h, k) is the vertex. Since 1 and 2 are where h and k are in vertex form, these are your vertex coordinates.
However, the original form is a negative number, so this means that in order for it to be + 1 then the 1 has to be negative (two negative make a positive). Therefore;
The vertex is (-1, 2).
In quadratic function in vertex form:
y = 5x² + 10x + 7
= 5(x² + 2x) + 7
= 5[(x² + 2x + 1) - 1] + 7
= 5(x² + 2x + 1) - 1*5 + 7
= 5(x + 1)² - 5 + 7
= 5(x + 1)² + 2.
The vertex of a quadratic function in vertex form is:
(-1, 2)
The vertex form should look like this: q(x - p)² + b
So the vertex is (p, b). That is why the vertex of 5(x + 1)² + 2 is (-1, 2).
