Problem 68 - Precalculus : Building Concepts and Connections

Question

Graph the function \(f(t)=t^{2}-4\) in a decimal window. Using your graph, determine the values of \(t\) for which \(f(t) \geq 0\).

Step-by-Step Solution

Answer

After graphing and analyzing the function \(f(t)=t^{2}-4\), it can be concluded that the values of \(t\) for which \(f(t) \geq 0\) are \(t \leq -2\) and \(t \geq 2\).

Step 1: Graphing the Function

The first task is graphing the function \(f(t)=t^{2}-4\). The vertex of the parabola, which is it's lowest point as it opens upward, can be found using the formula \(-\frac{b}{2a}\), where \(a\) and \(b\) are the coefficients of \(t^{2}\) and \(t\) respectively in the quadratic equation. The quadratic \(f(t) = t^{2} - 4\), however, doesn't have a \(t\) term, so \(b = 0\). We get the vertex to be at \(t = 0\). When \(t = 0\), \(f(t) = -4\). So the vertex of this parabola is at the point (0, -4).

Step 2: Finding the Roots

We must find which t-values make \(f(t) = 0\). To do this, we can solve the equation \(t^{2} - 4 = 0\). By solving, we get \(t = -2\) and \(t=2\). These two t-values are the roots of the quadratics and are also the x-intercepts of the function when graphed. This means that the parabola crosses the t-axis at t = -2 and 2.

Step 3: Determining Values for which \(f(t) \geq 0\)

The roots found earlier split the t-axis into three intervals: \(-\infty\) to -2, -2 to 2, and 2 to \(\infty\). For each interval, we can pick a test point and substitute it into the original function. If the function value is positive, then all values in that interval make \(f(t) \geq 0\). The test points -3, 0, 3 can be chosen for the three respective intervals. By substituting these points in the function \(f(t)=t^{2}-4\), we find that \(f(t) \geq 0\) when \(t \leq -2\) and \(t \geq 2\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a key feature when graphing quadratic functions. It represents either the highest or lowest point on the graph, known as the maximum or minimum, respectively. Given a quadratic function in the form of \( f(x) = ax^2 + bx + c \), the vertex can be calculated using the formula \( x = -\frac{b}{2a} \).

In our exercise with the function \( f(t) = t^2 - 4 \), there is no \( t \) coefficient, meaning \( b = 0 \). Therefore, the vertex is at \( t = 0 \). Substituting this into the function to solve for \( f(t) \), we find that the vertex is located at the point (0, -4). Understanding the vertex allows students to easily sketch the overall shape of the parabola and locate it correctly on the coordinate plane.

X-Intercepts of Quadratic

The x-intercepts of a quadratic function, often referred to as the 'roots', are the points where the graph crosses the x-axis. To find these points, we set the quadratic function equal to zero and solve for the variable.

In the context of our exercise, the quadratic function is \( f(t) = t^2 - 4 \), which factors into \( (t - 2)(t + 2) = 0 \). This yields two solutions, \( t = 2 \) and \( t = -2 \). Therefore, the parabola intersects the x-axis, or in our function's variable terms, the 't-axis', at these points. The x-intercepts provide crucial information about the symmetry of the parabola and are used to graphically represent solutions to the equation when set equal to zero.

Inequalities in Quadratic Functions

Understanding inequalities in the context of quadratic functions is essential for solving a wide range of problems, including those that ask for the set of values for which the quadratic function lies above or below a certain threshold. In mathematical terms, this involves determining where \( f(x) \) is greater than or less than a number.

In our exercise, after identifying the vertex and x-intercepts, we must determine for what values of \( t \), the function \( f(t) \) will be greater than or equal to zero. With the x-intercepts at \( t = -2 \) and \( t = 2 \), we observe that the parabola opens upwards. Therefore, the function will have values \( f(t) \geq 0 \) for \( t \leq -2 \) and \( t \geq 2 \). This approach entails testing intervals around the roots to see where the function produces positive outcomes, thus providing the solution to the inequality.