For the quadratic function f(x) = - x^2 +2x + 6, find the vertex and the axis of symmetry, and graph the function.
Again, I need help with this as these are practice examples for an upcoming quiz and I stink at this part of algebra and I need to get it otherwise I won't graduate and I have worked really hard at this, so please I need help with knowing how to do this, Please!!!
for F(x) = ax^2+bx+c, the vertex lies on the axis of symmetry, at x = -b/2a.
so, for your function, that is at
x = -2/-2 = 1
f(1) = 7
so, the vertex is at (1,7). See the graph at
http://www.wolframalpha.com/input/?i=-+x^2+%2B2x+%2B+6
Recall that the quadratic formula says that the roots of such a polynomial lie at
x = [-b±√(b^2-4ac)]/(2a)
That is just
x = -b/2a ±√(b^2-4ac)/2a
Note that axis of symmetry lies midway between the roots, at x = -b/2a
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!
I understand that quadratic functions can sometimes be challenging. Don't worry, I'll walk you through the steps to find the vertex and the axis of symmetry for the given quadratic function and help you graph it.
To find the vertex of a quadratic function in the form of f(x) = ax^2 + bx + c, you can use the formula:
x = -b / (2a)
y = f(x)
In this case, for the function f(x) = -x^2 + 2x + 6, we have a = -1, b = 2, and c = 6.
To find the x-coordinate of the vertex (axis of symmetry):
x = -b / (2a)
x = -2 / (2*(-1))
x = -2 / (-2)
x = 1
Now plug in this x-coordinate into the function to find the y-coordinate of the vertex:
y = -x^2 + 2x + 6
y = -(1)^2 + 2(1) + 6
y = -1 + 2 + 6
y = 7
So, the vertex of the quadratic function is (1, 7). The axis of symmetry is the vertical line through the vertex, which in this case is x = 1.
Now let's graph the function:
1. Plot the vertex at (1, 7).
2. Choose some x-values on either side of the vertex and calculate the corresponding y-values using the function.
For example, when x = 0, y = -(0)^2 + 2(0) + 6 = 6.
When x = 2, y = -(2)^2 + 2(2) + 6 = 4.
3. Plot these points on the graph.
4. Draw a smooth curve passing through the plotted points to represent the shape of the quadratic function.
The completed graph of the quadratic function f(x) = -x^2 + 2x + 6 should look like a downward-facing parabola with the vertex at (1, 7).
Remember to use graphing tools or graph paper to accurately plot and draw the graph. Good luck with your quiz preparation, and I hope this helps you better understand quadratic functions!