Question

SKETCHING GRAPHS Sketch the graph of the function. Label the vertex. $$ y=-\frac{1}{4} x^{2}-x-1 $$

Short answer

The graph of the function is a downward opening parabola with the vertex located at the point (2, -3).

Step-by-step solution

Identify the parameters for the quadratic function

First, identify the parameters in the given quadratic function: \(a=-\frac{1}{4}\), \(b=-1\), and \(c=-1\).

Determine the vertex of the function

Next, calculate the x-coordinate of the vertex using the formula: \(-\frac{b}{2a}\). Substituting the values into this formula, \[x = -\frac{-1}{2*-1/4} = 2\]. Then, substitute this x-value into the original equation to get the y-coordinate of the vertex: \[y = -\frac{1}{4}*2^2 - 2 - 1 = -3\]. So, the vertex of the function is (2, -3).

Sketching the function

Start by plotting the vertex at point (2, -3). Since the coefficient of \(x^2\) (\(a=-\frac{1}{4}\)) is negative, the graph opens downward. Sketch a parabola that opens downwards originating from the vertex at (2, -3), make sure that the parabola is symmetrically about the line \(x=2\). Label the vertex on the graph.

Key concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial in the realm of quadratic functions. The vertex is the highest or lowest point on the graph of a parabola, depending on whether it opens upwards or downwards. In mathematical terms, for a quadratic function of the form \(y=ax^2+bx+c\), the x-coordinate of the vertex can be found using the formula \(x = -\frac{b}{2a}\). Once you compute the x-coordinate of the vertex, you can substitute it back into the original equation to find the corresponding y-coordinate.

For the given function \(y=-\frac{1}{4}x^2-x-1\), we know that \(a=-\frac{1}{4}\), \(b=-1\), and \(c=-1\), suggesting that the graph will have a maximum point. Using the formula, we find the vertex's x-coordinate to be 2, and substituting this into the function gives us the y-coordinate, which is -3. Hence, the vertex for this parabola is at the point (2, -3), a vital piece of information for sketching the entire graph accurately.

Quadratic Functions

Quadratic functions form the backbone of many algebraic explorations and are represented by the standard form \(y=ax^2+bx+c\). Here, \(a\), \(b\), and \(c\) are constants, with \(a\) not equal to zero. The graph of a quadratic function is called a parabola. A pivotal feature of quadratic functions is their symmetry, with the axis of symmetry always passing through the vertex. The direction the parabola opens -- upward for \(a>0\) or downward for \(a<0\) -- is determined by the sign of coefficient \(a\).

If the coefficient \(a\) is positive, the parabola opens upwards, and the vertex represents the lowest point. Conversely, if \(a\) is negative, as in our exercise, the parabola opens downwards, and the vertex is the highest point. This affects the range of the function and, combined with the vertex, provides a clear starting point for graphing it.

Graphing Parabolas

Graphing parabolas is a visual way to comprehend the behavior of quadratic functions. When graphing, start by identifying the vertex; this point serves as the 'anchor' of your graph. For our example, the vertex (2, -3) acts as the peak since it opens downward. To sketch the parabola, plot the vertex, and then use the axis of symmetry, which is the vertical line that runs through the vertex, in this case, \(x=2\).

Now, because the coefficient \(a=-\frac{1}{4}\) affects the 'width' of the parabola making it more 'shallow' than the standard \(y=x^2\), draw the graph to reflect this by ensuring it broadens out more slowly than it would if \(a\) were 1. To achieve symmetry, plot equal x-values on both sides of the axis of symmetry and calculate their corresponding y-values. By connecting these points and smoothly curving through the vertex, a symmetric parabola that opens downward is formed. Labeling the key features like the vertex, axis of symmetry, and intercepts helps to make the graph more informative and easier to understand.