Which Of The Following Functions Is Graphed Below

Deciphering the Graph: Identifying the Underlying Function

Identifying the function represented by a graph requires a keen eye for detail and a solid understanding of fundamental mathematical concepts. This article will delve into the process of analyzing graphs to determine the underlying function, covering various function types and techniques to aid in accurate identification. We'll explore strategies for analyzing key features like intercepts, asymptotes, and the overall behavior of the graph, equipping you to confidently tackle such problems.

1. Analyzing Key Features of the Graph

Before we delve into specific function types, let's discuss the critical features we need to identify in any graph:

• Intercepts: Where does the graph intersect the x-axis (x-intercepts or roots) and the y-axis (y-intercept)? X-intercepts occur when y=0, and y-intercepts occur when x=0. These points provide valuable initial clues about the function. For example, a y-intercept of (0, 2) suggests a constant term of 2 in the function. Multiple x-intercepts hint at a polynomial function of a degree equal to or greater than the number of intercepts.

• Asymptotes: Does the graph approach any lines without ever touching them? These are asymptotes. Vertical asymptotes indicate values of x where the function is undefined (often resulting from division by zero). Horizontal asymptotes reveal the function's behavior as x approaches positive or negative infinity. The presence of asymptotes strongly suggests a rational function (a ratio of polynomials).

• Symmetry: Is the graph symmetric about the y-axis (even function), the origin (odd function), or neither? Even functions satisfy f(x) = f(-x), while odd functions satisfy f(-x) = -f(x). Symmetry can dramatically simplify the identification process.

• Turning Points (Extrema): How many times does the graph change from increasing to decreasing or vice versa? These turning points represent local maxima or minima of the function. The number of turning points provides information about the degree of a polynomial function (a polynomial of degree 'n' has at most 'n-1' turning points).

• Increasing/Decreasing Intervals: Over what intervals does the graph rise (increasing) and fall (decreasing)? This helps determine the overall behavior and potential function type. For example, a constantly increasing function might suggest an exponential function.

• Domain and Range: What are the possible input (x) and output (y) values? These limitations can rule out certain function types. A restricted domain (e.g., x > 0) could suggest a logarithmic or square root function.

2. Common Function Types and Their Graphical Characteristics

Now let's examine the graphical characteristics of several common function types:

• Linear Functions (f(x) = mx + b): These are straight lines with a constant slope (m) and y-intercept (b). They have no turning points, asymptotes, or symmetry (except for a horizontal line, where m=0).

• Quadratic Functions (f(x) = ax² + bx + c): These are parabolas. They have one turning point (a vertex), and their symmetry is always about a vertical line passing through the vertex. The sign of 'a' determines whether the parabola opens upwards (a > 0) or downwards (a < 0).

• Polynomial Functions (f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀): These functions have smooth, continuous curves. The highest power of x (n) is the degree of the polynomial, which determines the maximum number of turning points (n-1) and x-intercepts (n).

• Rational Functions (f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials): These functions often have asymptotes (vertical where Q(x) = 0 and horizontal depending on the degrees of P(x) and Q(x)). They may also have x-intercepts where P(x) = 0 and a y-intercept at P(0)/Q(0), provided Q(0) ≠ 0.

• Exponential Functions (f(x) = aˣ, where a > 0 and a ≠ 1): These functions exhibit exponential growth (a > 1) or decay (0 < a < 1). They have a horizontal asymptote at y = 0 if a is between 0 and 1. They are always positive.

• Logarithmic Functions (f(x) = logₐx, where a > 0 and a ≠ 1): These are the inverse functions of exponential functions. They have a vertical asymptote at x = 0 and are only defined for positive x values.

• Trigonometric Functions (sin x, cos x, tan x, etc.): These functions are periodic, meaning their graphs repeat over regular intervals. They have specific characteristics based on their amplitude, period, and phase shift.

3. Step-by-Step Approach to Identifying the Function

Let's outline a systematic approach to identifying the function represented by a given graph:

1. Examine the Intercepts: Determine the x-intercepts and y-intercept. This will give you immediate clues about the function.

2. Check for Asymptotes: Identify any vertical or horizontal asymptotes. Vertical asymptotes usually indicate rational functions, while horizontal asymptotes suggest exponential or rational functions.

3. Analyze Symmetry: Observe if the graph is symmetric about the y-axis, the origin, or neither. This determines if the function is even, odd, or neither.

4. Count the Turning Points: Note the number of turning points. For polynomials, this number is at most one less than the degree of the polynomial.

5. Determine Increasing/Decreasing Intervals: Identify the intervals where the function increases and decreases. This information will help you narrow down the possible function types.

6. Consider the Overall Shape: The overall shape of the graph provides valuable clues. Does it resemble a straight line (linear), a parabola (quadratic), an exponential curve, a logarithmic curve, or something else?

7. Use a Process of Elimination: Based on the features you've identified, eliminate function types that are inconsistent with the graph.

8. Estimate Parameters: If possible, estimate the parameters of the function (e.g., the slope and y-intercept for a linear function, the coefficient 'a' in a quadratic function).

9. Verify with Additional Points: Choose several points on the graph and check if they satisfy the function you've identified.

10. Use Technology: If you have access to graphing software or a calculator, you can use it to plot the function you've identified and visually compare it to the original graph.

4. Example: Identifying a Function from its Graph

Let's say we are presented with a graph exhibiting the following features:

• x-intercepts: x = -1, x = 2
• y-intercept: y = -2
• No asymptotes
• One turning point
• Opens upwards

Based on these features, we can conclude the following:

• No asymptotes eliminates rational functions.
• Two x-intercepts and one turning point strongly suggest a quadratic function (degree 2 polynomial).
• Opens upwards means the leading coefficient is positive.

The general form of a quadratic function is f(x) = a(x - r₁)(x - r₂), where r₁ and r₂ are the x-intercepts. In our case, r₁ = -1 and r₂ = 2. Therefore, the function is of the form f(x) = a(x + 1)(x - 2).

To find 'a', we use the y-intercept: f(0) = a(1)(-2) = -2a = -2. Solving for 'a' gives a = 1.

Therefore, the function is likely f(x) = (x + 1)(x - 2) = x² - x - 2. We can verify this by plotting the function and comparing it to the original graph.

5. Conclusion

Identifying the function represented by a graph is a skill developed through practice and careful observation. By systematically analyzing key features like intercepts, asymptotes, symmetry, and turning points, combined with a solid understanding of common function types and their graphical characteristics, you can confidently determine the underlying function. Remember to approach the problem methodically, using a process of elimination and verification to arrive at the correct answer. This detailed approach ensures accuracy and reinforces your understanding of function analysis.