Graph the function Identify the vertex and axis of symmetry

f(x)=2x^2 + 4x +1
please explain steps I need to do. I
keep looking at notes and just can't grasp any of it. Failing math not sure how to start again. I think I missed something major.

Well, failing math can be quite parabolling. But don't worry, I'm here to help! Let's break down the steps to graphing this function and finding its vertex and axis of symmetry.

Step 1: Identify the quadratic function as f(x) = 2x^2 + 4x + 1. This is a quadratic function because it has an x^2 term.

Step 2: To find the vertex, you can use the formula x = -b / (2a). In this case, a = 2 and b = 4. So, the x-value of the vertex is -4 / (2*2) = -4 / 4 = -1.

Step 3: To find the y-value of the vertex, plug the x-value (-1) back into the original function: f(-1) = 2(-1)^2 + 4(-1) + 1 = 2(1) - 4 + 1 = 2 - 4 + 1 = -1.

Step 4: Therefore, the vertex is (-1, -1).

Step 5: To find the axis of symmetry, it is simply the x-value of the vertex. So, the axis of symmetry is x = -1.

Step 6: Now, let's plot the vertex (-1, -1) on a graph. This point represents the lowest or highest point of the parabola, depending on whether the coefficient of x^2 is positive or negative.

Step 7: To further graph the parabola, you can choose a few more points on either side of the vertex. You can plug in some x-values into the original function and calculate the corresponding y-values.

Step 8: Once you have a couple more points, plot them on the graph, and then connect them smoothly to create the parabolic shape.

In summary, to graph the function f(x) = 2x^2 + 4x + 1, you need to find the vertex, which is (-1, -1), and the axis of symmetry, which is x = -1. I hope this explanation adds some positive value to your math understanding!

To graph the function and identify the vertex and axis of symmetry, you can follow these steps:

Step 1: Start by writing down the given function:
f(x) = 2x^2 + 4x + 1

Step 2: To find the vertex form of the function, complete the square. Begin by factoring out the coefficient of x^2 (which is 2 in this case):
f(x) = 2(x^2 + 2x) + 1

Step 3: Take half of the coefficient of x (which is 2 in this case) and square it. Add the result to the expression inside the parentheses.
f(x) = 2(x^2 + 2x + 1) - 2 + 1

Step 4: Simplify the expression inside the parentheses and combine like terms outside the parentheses:
f(x) = 2(x + 1)^2 - 1

Step 5: The vertex form of the quadratic function is f(x) = a(x - h)^2 + k, where (h, k) represents the vertex.
In our case, the vertex is (-1, -1).

Step 6: The axis of symmetry is a vertical line that passes through the vertex. In this case, the equation of the axis of symmetry is x = -1.

Step 7: The graph of the quadratic function is a parabola that opens upward if a > 0, as is the case here with a = 2.

Step 8: Use the vertex and axis of symmetry to plot the graph. Start by plotting the vertex (-1, -1). Then, find two more points on each side of the vertex and connect them smoothly to form a parabolic curve.

Step 9: To find additional points, you can choose x-values on both sides of the vertex (for example, x = -2, -3, 0, 1, 2), substitute them into the equation f(x), and calculate the corresponding y-values.

Step 10: Continue plotting these points on the graph and connect them to create the parabolic curve.

By following these steps, you should be able to graph the function, identify the vertex, and determine the axis of symmetry.

I'm sorry to hear that you're struggling with math, but I'm here to help you understand how to graph the function.

To find the vertex and axis of symmetry of a quadratic function, you'll need to follow these steps:

Step 1: Start with the given quadratic function, which is f(x) = 2x^2 + 4x + 1.

Step 2: Rewrite the function in the standard form ax^2 + bx + c. In this case, we already have it in standard form.

Step 3: Identify the values of a, b, and c from the standard form. In our function, a = 2, b = 4, and c = 1.

Step 4: Calculate the x-coordinate of the vertex using the formula x = -b/2a. In our case, x = -4 / (2 * 2) = -1.

Step 5: Substitute the found x-coordinate (-1) back into the function to find the y-coordinate of the vertex. f(-1) = 2(-1)^2 + 4(-1) + 1 = -1.

Step 6: The vertex of the parabola is given by the coordinates (x, y), where x is the x-coordinate and y is the y-coordinate of the vertex. In our case, the vertex is (-1, -1).

Step 7: The axis of symmetry is a vertical line that passes through the vertex of the parabola. The equation of the axis of symmetry is x = x-coordinate of the vertex. So, in our case, the axis of symmetry is x = -1.

Now that we have found the vertex (-1, -1) and the axis of symmetry (x = -1), we can proceed to graph the function. The vertex represents the lowest point (in this case, since the coefficient of x^2 is positive), and the axis of symmetry divides the parabola into two symmetric halves.

To plot the graph, start by drawing the axis of symmetry (a vertical line at x = -1). Then, plot a few points on each side of the axis and connect them smoothly to form a parabolic shape. You can choose various x-values and calculate the corresponding y-values using the function f(x) = 2x^2 + 4x + 1.

Once you have plotted enough points, you should see a symmetrical U-shaped curve extending downwards with the vertex at (-1, -1).

I hope this explanation helps!

as with any quadratic ax^2+bx+c, the vertex lies on the axis of symmetry, at x = -b/2a

In this case, that is x = -1
f(-1) = -1
so the vertex is at (-1,-1)

Or, you can complete the square to find the vertex:

f(x) = 2x^2+4x+1
= 2(x^2+2x)+1
= 2(x^2+2x+1)+1-2
= 2(x+1)^2 - 1

Recall that if y = a(x-h)^2+k then the vertex is at (h,k)

Here, it is clear that (h,k) = (-1,-1)