Problem 48 - Algebra 2

Question

CHALLENGE A quadratic function has values \(f(-4)=-11, f(-2)=9\) , and \(f(0)=5 .\) Between which two \(x\) -values must \(f(x)\) have a zero? Explain your reasoning.

Step-by-Step Solution

Answer

The function has a zero between \( x = -4 \) and \( x = -2 \) because the sign changes between these values.

Step 1: Understand the Problem

We need to determine between which two values of \( x \) the quadratic function \( f(x) \) changes its sign (crosses the x-axis), which is indicative of having a zero.

Step 2: Evaluate the given function values

The function values provided are \( f(-4) = -11 \), \( f(-2) = 9 \), and \( f(0) = 5 \).

Step 3: Consider the function change from f(-4) to f(-2)

From \( f(-4) = -11 \) to \( f(-2) = 9 \), the function value changes from negative to positive. As the function changes from negative to positive, it crosses the x-axis between these two points.

Step 4: Consider the function change from f(-2) to f(0)

From \( f(-2) = 9 \) to \( f(0) = 5 \), the function remains positive. Therefore, there is no zero of the function between these two points.

Step 5: Conclusion

Since there is a sign change between \( x = -4 \) and \( x = -2 \), it indicates that there is at least one zero between these points.

Key Concepts

Zeros of a Function

In quadratic functions, the zeros are the values of \( x \) for which \( f(x) = 0 \). These zeros are the points where the graph of the function cuts the x-axis.
For a given quadratic function \( f(x) \), finding these zeros is crucial because they help us understand the behavior of the graph. Zeros can be found by solving the equation \( f(x) = 0 \).
In our exercise, we are looking for zeros between specific \( x \) values, indicating points where \( f(x) \) changes from negative to positive or vice versa.

Sign Change

A sign change in a function's values indicates a potential zero in its graph. This happens when \( f(x) \) transitions from positive to negative, or from negative to positive.
In mathematical terms, a sign change implies that the function has moved from one side of the x-axis to the other.

• From \( f(-4) = -11 \) to \( f(-2) = 9 \), there's a change in the sign, moving from negative to positive.
• This sign change reveals the function crosses the x-axis between these \( x \) values, implying the presence of a zero.

This insight is essential in efficiently finding zeros in a function without graphing the entire curve.

Function Values

The values of \( f(x) \) at different points offer significant clues about the behavior and topology of the function.
In the context of our quadratic function, the given values \( f(-4) = -11 \), \( f(-2) = 9 \), and \( f(0) = 5 \) provide a snapshot of how \( f(x) \) behaves over the x-axis.

• These values point out where significant changes occur, for example, when the function value moves from negative to positive, which is a sure sign of crossing the x-axis.
• Observing these values helps predict zeros and analyze crossing patterns that are critical for understanding the entire function.

Crossing the X-Axis

When a quadratic function crosses the x-axis, it represents a root or zero of the function. This crossing signifies an \( x \) value where \( f(x) = 0 \).
Identifying these crossings is key to understanding quadratic functions because they represent solutions to the equation \( f(x) = 0 \).

• In our challenge, the function crosses the x-axis between \( x = -4 \) and \( x = -2 \).
• This crossing is confirmed by a sign change from negative to positive, indicating a zero exists in this interval.

Thus, monitoring these crossings allows us to pinpoint where zeros occur without needing a complete graph or complex calculations.