Problem 90 - Algebra and Trigonometry Real Mathematics, Real People

Question

Graph the function and determine the interval(s) (if any) on the real axis for which \(f(x) \geq 0\) Use a graphing utility to verify your results. $$f(x)=x^{2}-4 x$$

Step-by-Step Solution

Answer

The intervals of x where the function \(f(x) = x^2 - 4x\) is nonnegative (\(f(x) \geq 0\)) are \(x \leq 0\) and \(x \geq 4\).

Step 1: Identify and Plot the Zero Points

The zero points of the function, also known as the roots or the solution of equation, are obtained by setting \(f(x) = 0\). Hence, \(x^2 - 4x = 0\). By factoring, we obtain \(x * (x - 4) = 0\). This gives us \(x = 0\) or \(x = 4\) as the zero points of the function.

Step 2: Calculate the Vertex of the Parabola

The vertex form of a parabola is \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola. The x-coordinate of the vertex \(h\) can be found using the formula \(h = -b/2a\). Substituting \(a = 1\) and \(b = -4\) into the formula gives us \(h = -(-4)/(2*1) = 2\). Substituting \(x = 2\) into \(f(x)\) gives us the y-coordinate of the vertex \(f(2) = 2^2 - 4*2 = -4\). Hence, the vertex of the parabola is at \((2,-4)\).

Step 3: Determine the Interval where the Function is Nonnegative

From the parabola and its vertex, we notice that the graph starts at \(x=0\) (y=0), goes down to \(x=2\) (y=-4), and then rises up after \(x=4\) (y=0), which means the function is greater than or equal to 0 for \(x \leq 0\) and \(x \geq 4\). Therefore, the intervals of x where \(f(x) \geq 0\) are \(-\infty < x \leq 0\) and \(4 \leq x < \infty\).

Step 4: Verification using a Graphing Utility

Although not shown here, plotting the graph using a graphing utility or calculator can verify these findings. If done correctly, the graph should show a concave upward parabola that touches the x-axis at \(x=0\) and \(x=4\), with a vertex at (2,-4). The function is nonnegative (above or on the x-axis) when \(x \leq 0\) and \(x \geq 4\) which confirms our solution.

Key Concepts

Graphing Parabolas

When graphing a quadratic function, the resulting graph is called a parabola. Parabolas are symmetric U-shaped curves that can open upwards or downwards. To graph a parabola, you need to understand its key features:

1. **Direction of Opening**: This depends on the coefficient of the squared term. If the coefficient (\(a\) in \(ax^2\)) is positive, the parabola opens upwards. If it is negative, it opens downwards.
2. **Vertex**: The highest or lowest point of the parabola, depending on whether it opens upwards or downwards.
3. **Axis of Symmetry**: A vertical line that goes through the vertex, dividing the parabola into two mirror-image halves.
4. **Roots or Zero Points**: Points where the parabola intersects the x-axis. These are also called x-intercepts or solutions to the quadratic equation.

For the function \(f(x) = x^2 - 4x\), the parabola opens upwards because the coefficient of \(x^2\) is positive (\(a=1\)). This knowledge is crucial when sketching the graph of the function.

Finding Intervals

Finding intervals where a function is above, below, or equal to zero involves analyzing its graph:

1. **Zero Points**: Solve the equation \(f(x) = 0\) to find where the function crosses the x-axis. These are your boundary points. For \(f(x) = x^2 - 4x\), the zero points are solutions of \(x^2 - 4x = 0\), which are \(x = 0\) and \(x = 4\).
2. **Test Intervals**: Use these zero points to define intervals on the x-axis and test whether the function is above or below the x-axis in these intervals. For \( f(x) = x^2 - 4x \), the intervals formed by the zero points are \((-\infty, 0)\), \((0, 4)\), and \((4, \infty)\).

By graphing or using a sign test, you can determine where the function is positive. In this example, you would find \(f(x)\) is greater than or equal to zero in the intervals \((-\infty, 0]\) and \([4, \infty)\). This analysis helps in understanding the behavior of the function across different ranges.

Vertex of a Parabola

The vertex is a crucial part of understanding a parabola's structure and the easiest way to find it is using the formula \( h = -\frac{b}{2a} \). This formula gives you the x-coordinate of the vertex for a standard quadratic function \( ax^2 + bx + c \).

Once you have the x-coordinate, substitute it back into the function to find the y-coordinate.

For the function \(f(x) = x^2 - 4x\):
\[ h = -\frac{-4}{2 \times 1} = 2 \]
Substitute \(x = 2\) into the function to get \( f(2) = 2^2 - 4 \times 2 = -4 \). Hence, the vertex is located at \((2, -4)\).

The vertex can be a maximum or a minimum point. Here, since the parabola opens upwards, the vertex at \((2, -4)\) is the minimum point on the graph. Understanding where the vertex lies will help you sketch the parabola accurately.

Nonnegative Intervals

Nonnegative intervals are parts of the x-axis where the function value is zero or positive. These intervals are where the parabola is either tangent to or above the x-axis.

Finding these intervals involves determining where the parabola crosses the x-axis, and analyzing sections of the graph in between and beyond these points.

For \( f(x) = x^2 - 4x \):

• Solving \( f(x) = 0 \) gave us the x-intercepts: \( x = 0 \) and \( x = 4 \).
• Since the vertex is at \((2, -4)\) below the x-axis, any part of the parabola opening upwards shoots outwards after \( x = 4 \).
• The function is nonnegative in the intervals where the graph touches or stays above the x-axis, occurring at \(( -\infty, 0 ]\) and \([4, \infty)\).

These intervals can be confirmed using graphing tools, showing that the parabola does not dip below the x-axis beyond these points.