find the vertex and axis of symmetry for
y = x^2 - 6x + 7
To find the vertex and axis of symmetry for the quadratic equation y = x^2 - 6x + 7, you can follow these steps:
Step 1: Identify the coefficient values in the equation. In this case, the coefficient of x^2 is 1, the coefficient of x is -6, and the constant term is 7.
Step 2: Use the formula for the x-coordinate of the vertex. The x-coordinate of the vertex (h) can be found using the formula h = -b / (2a), where a is the coefficient of x^2 and b is the coefficient of x.
In this case, a = 1 and b = -6. Plugging these values into the formula, we get:
h = -(-6) / (2 * 1)
= 6 / 2
= 3
Step 3: Find the y-coordinate of the vertex by substituting the x-coordinate (h) into the equation. Using h = 3, we can calculate the value of y:
y = (3)^2 - 6(3) + 7
= 9 - 18 + 7
= -2
Therefore, the vertex is (3, -2).
Step 4: Identify the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex. Its equation is x = h. In this case, x = 3.
So, the vertex is (3, -2) and the axis of symmetry is x = 3.
To find the vertex and axis of symmetry of a quadratic function in the form y = ax^2 + bx + c, you can use the formula:
x = -b / (2a)
Let's apply this formula to the given quadratic function: y = x^2 - 6x + 7.
Step 1: Identify the values of a, b, and c from the equation:
a = 1
b = -6
c = 7
Step 2: Use the formula x = -b / (2a) to find the x-coordinate of the vertex:
x = -(-6) / (2 * 1)
x = 6 / 2
x = 3
Step 3: Substitute the x-coordinate of the vertex into the original equation to find the y-coordinate:
y = (3)^2 - 6(3) + 7
y = 9 - 18 + 7
y = -2
Therefore, the vertex of the quadratic function y = x^2 - 6x + 7 is (3, -2).
Step 4: Find the axis of symmetry by simplifying the x-coordinate of the vertex:
Axis of symmetry: x = 3
So, the axis of symmetry for the given quadratic function is x = 3.