Write the quadratic function in standard form (if necessary) and sketch its graph.
Identify the vertex.
f(x)=x^2+3x+1/4
y = x^2 + 3x + 1/4
= x^2 + 3x + (3/2)^2 + 1/4 - (3/2)^2
= (x + 3/2)^2 - 2
That should answer most of your questions.
To write the quadratic function in standard form and identify the vertex, we can follow a few steps:
Step 1: Write the quadratic function in standard form.
Standard form of a quadratic function is given by:
f(x) = ax^2 + bx + c
Given function:
f(x) = x^2 + 3x + 1/4
Since the given function is already in standard form, we don't need to make any changes.
Step 2: Identify the coefficients a, b, and c.
From the standard form, we can identify the coefficients as:
a = 1, b = 3, c = 1/4
Step 3: Find the vertex using the formula:
The x-coordinate of the vertex, denoted as h, can be found using the formula:
h = -b / (2a)
In this case:
a = 1, b = 3
h = -3 / (2 * 1)
h = -3/2
Substitute the value of h back into the quadratic function to find the y-coordinate of the vertex.
f(h) = f(-3/2) = (-3/2)^2 + 3(-3/2) + 1/4
Simplifying the expression, we get:
f(-3/2) = 9/4 - 9/2 + 1/4
f(-3/2) = 9/4 - 18/4 + 1/4
f(-3/2) = -8/4
f(-3/2) = -2
So, the vertex of the quadratic function is at (-3/2, -2).
Step 4: Sketch the graph.
To sketch the graph, plot the vertex at (-3/2, -2) and the corresponding y-intercept at (0, 1/4). Also, choose a few more x-values (such as -4, -2, 0, 2, 4) and find the corresponding y-values by substituting those x-values into the quadratic function. Plot those points, and finally, connect the points to get a smooth curve.
Here's a rough sketch of the graph:
```
|
-4 | +
|
|
---------------------
|
-2 | +
|
|
---------------------
|
0 |
|
|
---------------------
|
2 | +
|
|
---------------------
|
4 | +
|
|
```
The vertex is the lowest point on the graph. In this case, the vertex is at (-3/2, -2).