Use the method of completing the square to find the standard form of the quadratic function. state the vertex and axis of symmetry of the graph of the function and then sketch the graph.
(a) f(x)= x^2+4x+1
To complete the square for the quadratic function f(x) = x^2 + 4x + 1, we follow these steps:
Step 1: Move the constant term to the right side:
f(x) - 1 = x^2 + 4x
Step 2: Take half of the coefficient of x, square it, and add it to both sides of the equation:
f(x) - 1 + (4/2)^2 = x^2 + 4x + (4/2)^2
f(x) - 1 + 4 = x^2 + 4x + 4
Step 3: Simplify both sides:
f(x) + 3 = (x + 2)^2
Step 4: Write the function in standard form:
f(x) = (x + 2)^2 - 3
The standard form of the quadratic function is f(x) = (x + 2)^2 - 3.
The vertex of the parabola is (-2, -3) and the axis of symmetry is x = -2.
To sketch the graph, we can start by plotting the vertex at (-2, -3). Then, we can find several other points by substituting different x-values into the equation. For example, when x = -4, y = (-4 + 2)^2 - 3 = 9. Similarly, when x = 0, y = (0 + 2)^2 - 3 = 1. We can now plot these points and connect them to form the graph of the quadratic function.
Here is a rough sketch of the graph:
```
|
| .
| .
-3 | .
|
|_____________________
-5 -4 -3 -2 -1 0 1 2 3 4 5
```
Note: The exact shape of the parabola depends on the scale of the axes. The sketch provided should give a general idea of what the graph looks like.
To find the standard form of the quadratic function using the method of completing the square, follow these steps:
Step 1: Start with the given quadratic function:
f(x) = x^2 + 4x + 1
Step 2: Group the x^2 and x terms:
f(x) = (x^2 + 4x) + 1
Step 3: Take half of the coefficient of the x-term (4 in this case) and square it:
(half of 4 is 2, and 2 squared is 4)
Step 4: Add the result from Step 3 to both sides of the equation:
f(x) + 4 = (x^2 + 4x + 4) + 1
Step 5: Rewrite the trinomial as a perfect square:
f(x) + 4 = (x + 2)^2 + 1
Step 6: Simplify the equation:
f(x) = (x + 2)^2 - 3
So, the standard form of the quadratic function is:
f(x) = (x + 2)^2 - 3
The vertex of the graph is (-2, -3) and the axis of symmetry is x = -2.
To sketch the graph, start by plotting the vertex (-2, -3). Then, determine a few other points by substituting different x-values into the function. Once you have enough points, connect them with a smooth curve. The graph should be a U-shaped curve that opens upwards, with the vertex as the lowest point.