I understand that you need help with finding the vertex and axis of symmetry for a quadratic function, as well as graphing it. I'll be happy to guide you through the steps.
First, let's find the vertex and axis of symmetry.
Step 1: To find the x-coordinate of the vertex, use the formula x = -b / (2a), where a, b, and c are the coefficients from the quadratic function in the form ax^2 + bx + c.
In this case, the quadratic function is f(x) = - x^2 + 2x + 6. Comparing it with the form ax^2 + bx + c, we have a = -1 and b = 2.
Using the formula x = -b / (2a), we can substitute the values:
x = -(2) / (2*(-1))
x = -2 / -2
x = 1
Therefore, the x-coordinate of the vertex is 1.
Step 2: To find the y-coordinate of the vertex, substitute the x-coordinate we found (1) back into the equation f(x).
f(1) = - (1)^2 + 2(1) + 6
f(1) = -1 + 2 + 6
f(1) = 7
So, the y-coordinate of the vertex is 7.
Therefore, the vertex is (1, 7).
Step 3: To find the axis of symmetry, simply use the x-coordinate of the vertex. In this case, the axis of symmetry is x = 1.
Now, let's move on to graphing the function.
Step 4: Plot the vertex (1, 7) on a coordinate grid.
Step 5: Since the coefficient of the x^2 term is -1, the parabola will open downwards. This means it will be a "U" shape.
Step 6: Find a few additional points on the graph by substituting different x-values into the equation f(x) and calculating the corresponding y-values. For example, you can choose x = 0, -1, and 2.
f(0) = - (0)^2 + 2(0) + 6
f(0) = 0 + 0 + 6
f(0) = 6
f(-1) = - (-1)^2 + 2(-1) + 6
f(-1) = -1 - 2 + 6
f(-1) = 3
f(2) = - (2)^2 + 2(2) + 6
f(2) = -4 + 4 + 6
f(2) = 6
Step 7: Plot the additional points (0, 6), (-1, 3), and (2, 6) on the graph.
Step 8: Connect the plotted points and draw a smooth curve that passes through them, creating the parabolic shape.
Step 9: Lastly, label the graph with the axis of symmetry (x = 1), vertex (1, 7), and any other relevant information.
I hope this step-by-step explanation helps you understand how to find the vertex and axis of symmetry for a quadratic function and graph it. If you have any more questions, feel free to ask!