Problem 7 - Algebra and Trigonometry

Question

\(5-12\) . A function \(f\) is given. (a) Use a graphing device to draw the graph of \(f\) . (b) State approximately the intervals on which \(f\) is increasing and on which \(f\) is decreasing. $$ f(x)=x^{2}-5 x $$

Step-by-Step Solution

Answer

The function is decreasing on \((-\infty, 2.5]\) and increasing on \([2.5, \infty)\).

Step 1: Understand the Function

The function given is \( f(x) = x^2 - 5x \). It is a quadratic function, which generally forms a parabola when graphed. The coefficient of \( x^2 \) is positive, indicating the parabola opens upwards.

Step 2: Graph the Function

Use a graphing device or software (such as Desmos, a graphing calculator, or other online tools) to visualize the function \( f(x) = x^2 - 5x \). You will see the shape of the parabola and identify specific points, such as the vertex.

Step 3: Find the Vertex of the Parabola

The vertex form of a quadratic function is helpful to determine the peak or trough. The vertex for a standard form \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). For \( f(x) = x^2 - 5x \), \( a = 1 \) and \( b = -5 \), so the vertex \( x \)-coordinate is \( x = -\frac{-5}{2(1)} = \frac{5}{2} = 2.5 \). Substitute back into the function to find the y-coordinate, thus the vertex is at \( \left(2.5, f(2.5)\right) \).

Step 4: Derive the Function

To determine where the function is increasing or decreasing, find its derivative. The derivative of \( f(x) = x^2 - 5x \) is \( f'(x) = 2x - 5 \). This tells us the slope of the tangent to the curve at any point \( x \).

Step 5: Determine Critical Points

Set \( f'(x) = 0 \) to find the critical points: \( 2x - 5 = 0 \) leads to \( x = 2.5 \). This critical point divides the graph into regions to check intervals of increase and decrease.

Step 6: Test Intervals Around Critical Points

Evaluate \( f'(x) \) on intervals around the critical point to determine whether \( f(x) \) is increasing or decreasing. For \( x < 2.5 \), choose a test point like \( x = 0 \), \( f'(0) = -5 \) (negative), so \( f(x) \) is decreasing. For \( x > 2.5 \), choose \( x = 3 \), \( f'(3) = 1 \) (positive), so \( f(x) \) is increasing.

Step 7: State Intervals of Increase and Decrease

From the derivative test, the function \( f(x) \) is decreasing on \( (-\infty, 2.5] \) and increasing on \( [2.5, \infty) \).

Key Concepts

Graphing Parabolas

A quadratic function like \( f(x) = x^2 - 5x \) typically forms a parabola when graphed. Recognizing this shape is crucial for understanding the function's behavior.

• The coefficient of the \( x^2 \) term determines the parabola's orientation. If positive, as in \( f(x) = x^2 - 5x \), the parabola opens upwards.
• Using a graphing tool can help you visualize the parabola and identify features like the vertex and axis of symmetry.

To graph the parabola yourself:1. **Identify the Vertex**: Calculate the vertex \( x \)-coordinate using \( x = -\frac{b}{2a} \). For our function, it’s \( x = 2.5 \). Plug this back into the function to find the \( y \)-coordinate.2. **Plot the Vertex**: With coordinates \( (2.5, f(2.5)) \), this point on the graph represents the turning point or the lowest point on the parabola since it opens upwards.3. **Draw the Symmetry**: The line \( x = 2.5 \) is the axis of symmetry, placing the vertex at the center of the parabola. Drawing points equidistant from this line will help complete the graph. Graphing the parabola manually or with technology reveals not just shape, but behavior, making this a foundational step in studying quadratic functions.

Increase and Decrease Intervals

Understanding where a function increases or decreases helps predict its behavior. For the function \( f(x) = x^2 - 5x \), intervals of increase and decrease tell us where the function’s output is rising or falling as \( x \) changes. This has practical applications in many scenarios for predicting future trends or behaviors in practical applications.

• **Find the Derivative**: The first step is to calculate the derivative, \( f'(x) = 2x - 5 \), which reveals changes in the function’s slope.
• **Identify Critical Points**: Set \( f'(x) = 0 \) to find critical points. For this function, it occurs at \( x = 2.5 \), indicating potential switches from increasing to decreasing or vice versa.
• **Determine Intervals**: Use test points on either side of \( x = 2.5 \) to identify function behavior.
  • For \( x < 2.5 \), \( f'(x) \) is negative (e.g., \( x = 0 \)), indicating a decrease.
  • For \( x > 2.5 \), \( f'(x) \) is positive (e.g., \( x = 3 \)), indicating an increase.

Thus, we conclude the function decreases on \((-\infty, 2.5]\) and increases on \([2.5, \infty)\). Grasping these concepts allows students to picture how changes in \( x \) affect the overall graph behavior.

Derivative

The derivative is a powerful tool that provides insights into the function's behavior. It not only helps in identifying slopes but also aids in understanding trends of increasing or decreasing areas.

• **Definition**: The derivative, \( f'(x) \), is a formula that describes how the function value \( f(x) \) changes as the input \( x \) changes.
• **Application**: By finding \( f'(x) \) of \( f(x) = x^2 - 5x \), we derive \( f'(x) = 2x - 5 \). This tells us the rate of change of the function and can mean a lot about the function's increasing or decreasing nature.
• **Critical Point Analysis**: Setting \( f'(x) = 0 \) enables us to find critical points, such as \( x = 2.5 \) for our function. This could signify the transition between increasing and decreasing intervals.

By understanding how to derive functions and interpret the results, the derivative becomes a powerful tool in predicting and analyzing the behavior of quadratic functions and many other types of functions students may encounter.