Question

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$f(t)=\frac{1}{2}\left(t^{2}-4 t-1\right)$$

Short answer

The function has a y-intercept at \(-\frac{1}{2}\), roots can be computed by solving \(t^2 - 4t - 1= 0\), and a point of extremity at \(t=2\). The graph is a upward facing parabola. By drawing points based on this information and joining them in the pattern of an upward facing parabola, we can sketch the graph by hand. The sketch can be further confirmed by using a graphing utility.

Step-by-step solution

Determine the y-intercept

To find the y-intercept of the function, set \(t=0\) into the function, resulting in \(f(0) = \frac{1}{2}(0^2 - 4*0 - 1) = -\frac{1}{2}\). Therefore, the y-intercept of the function is \(-\frac{1}{2}\).

Compute the zeros of the function

To find the roots of the function, set it equal to zero: \(\frac{1}{2}(t^2 - 4t - 1) = 0\). This can be rewritten as \(t^2 - 4t - 1 = 0\). Solving this quadratic equation using the quadratic formula \(t=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\), we get two roots.

Compute the derivative and find the points of extremity

Taking the derivative of the function gives us \(f'(t) = t - 2\). To find points of extremity, set this equal to zero and solve for \(t\). Doing so, we find one point of extremity at \(t=2\). Use the second derivative or plugging points around \(2\) into the first derivative to classify this point as a maximum or minimum.

Sketching the graph

With the knowledge of the shape of the graph from the type of the function (quadratic function which opens upwards), the y-intercept, roots and points of extremity, sketch the graph by hand.

Confirming using a graphing utility

Finally, confirm the correctness of your sketch by using a graphing tool to plot the function. It should match up with the sketch.

Key concepts

Y-Intercept Calculation

Understanding where a function crosses the y-axis is crucial for graph analysis, as it provides a starting point for sketching the graph. The y-intercept is found by setting the variable representing the horizontal axis, often 'x' or 't' in algebra, to zero and solving the function for that value. In our exercise, for the function

\( f(t) = \frac{1}{2}(t^2 - 4t - 1) \),

we set \( t = 0 \) which simplifies the function to

\( f(0) = -\frac{1}{2} \).

Thus, the y-intercept of the graph is the point (0, -0.5). This provides a reference point on the graph and is essential for sketching the function accurately.

Finding Roots of Functions

The roots of a function are the points where the graph crosses the horizontal axis. To find these intersections, we set the function equal to zero. For the quadratic function

\( f(t) = \frac{1}{2}(t^2 - 4t - 1) \),

we solve the equation

\( t^2 - 4t - 1 = 0 \).

Using the quadratic formula

\( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \),

yields the roots of the function. These roots are essential for graph sketching as they mark the points where the function's curve will intersect the t-axis.

Derivative Computation

The derivative of a function provides information on its slope at any given point, indicating how the function is changing. It's instrumental in finding extremities which are the peaks and troughs on the graph. For our function,

\( f'(t) = t - 2 \),

where the prime denotes the derivative. Setting the derivative equal to zero and solving for \( t \) gives us the extremity points. In this example, \( t = 2 \) signifies a point where there might be a peak or a trough in the graph, which contributes to our understanding of the function’s shape and behavior.

Sketching Quadratic Graphs

Quadratic graphs are parabolic, and their general shape is either a 'smile' or a 'frown', based on the coefficient of the \( t^2 \) term. With the function given,

\( f(t) = \frac{1}{2}(t^2 - 4t - 1) \),

the positive coefficient of the \( t^2 \) term indicates the graph opens upwards like a 'smile'. Combining the y-intercept, roots, and extremity points obtained from our previous computations allows us to sketch the basic outline of this parabolic graph by hand. For higher accuracy, plotting additional points or applying methods such as vertex form can be beneficial.

Graphing Utility Confirmation

After sketching the function manually, confirming its accuracy with the use of a graphing utility is an important step. Graphing calculators or computer software plot functions with high precision, giving a reliable visual representation of the graph. By comparing the sketched graph to the computer-generated one for

\( f(t) = \frac{1}{2}(t^2 - 4t - 1) \),

any discrepancies can be identified and rectified. This reinforces understanding of graphical features and ensures that the hand-drawn graph is a true reflection of the function's behavior.