# Step-by-Step Solution for Rational Function Construction Below is a **general guide** for constructing a rational function from a graph. Since you did not provide a specific graph, I’ll outline the exact steps and answer format you should use for any similar problem. If you provide specific intercepts or asymptotes, I can fill those in. --- ## A. Vertical Asymptotes ### Step 1: Identify Vertical Asymptotes - **Observation:** Vertical asymptotes occur where the function is undefined (denominator is zero, but numerator is not). - **Equation:** The equations are of the form \( x = a \). ### Step 2: Corresponding Factor - **Factor:** Each vertical asymptote at \( x = a \) corresponds to a factor \( (x - a) \) in the **denominator**. ### Step 3: Where Should Factor Appear? - **Location:** These factors must be in the **denominator** of the rational function. **Example Table:** | Vertical Asymptote | Factor | Numerator/Denominator | |--------------------|-------------|-----------------------| | \( x = a \) | \( x-a \) | Denominator | | \( x = b \) | \( x-b \) | Denominator | --- ## B. X-intercepts ### Step 1: Identify X-intercepts - **Observation:** X-intercepts are where the graph crosses or touches the x-axis. - **Value:** These are points where \( y = 0 \). ### Step 2: Crossing or Touching - **Cross:** If the graph crosses the x-axis at \( x = c \), the multiplicity is **odd**. - **Touch:** If the graph touches (bounces off) the x-axis at \( x = c \), the multiplicity is **even**. ### Step 3: Corresponding Factor - **Factor:** Each x-intercept at \( x = c \) corresponds to a factor \( (x - c) \) in the **numerator**. ### Step 4: Even or Odd Power - **Odd power:** Crosses the axis. - **Even power:** Touches the axis. ### Step 5: Where Should Factor Appear? - **Location:** These factors must be in the **numerator** of the rational function. **Example Table:** | X-intercept | Cross/Touch | Factor | Even/Odd Power | Numerator/Denominator | |-------------|-------------|-------------|----------------|-----------------------| | \( x = c \) | Cross | \( x-c \) | Odd | Numerator | | \( x = d \) | Touch | \( x-d \) | Even | Numerator | --- ## C. Forming a Trial Function ### Step 1: Combine Factors - **Numerator:** Product of all factors for x-intercepts, each raised to the correct power. - **Denominator:** Product of all factors for vertical asymptotes, each raised to the correct power. ### Step 2: Write Trial Function \[ f(x) = \frac{(x-c_1)^{p_1} (x-c_2)^{p_2} \dots}{(x-a_1)^{q_1} (x-a_2)^{q_2} \dots} \] - Where \( c_i \) are x-intercepts, \( p_i \) their powers (even/odd). - \( a_j \) are vertical asymptotes, \( q_j \) their multiplicities. ### Step 3: Use Calculator/Graphing Tool - **Graph the trial function.** - **Check:** Does the graph match? If not, you may need to: - Change a power (even ↔ odd). - Multiply the numerator by \(-1\) to reflect across the x-axis. ### Step 4: Check y-intercept - Plug \( x = 0 \) into your function. - Compare with the y-intercept from the given graph. - **If not matching:** Multiply the entire function by a constant \( k \) to adjust the y-intercept. ### Step 5: Check Horizontal Asymptote - **Degree test:** - If numerator degree < denominator degree: Asymptote at \( y = 0 \). - If degrees are equal: Asymptote at \( y = \frac{\text{leading coefficient numerator}}{\text{leading coefficient denominator}} \). - If numerator degree > denominator degree: No horizontal asymptote (may be oblique). - **If they do not match:** Adjust the leading coefficients or degrees accordingly. --- ## D. Write Final Function - **Final Answer:** Write the function with all necessary adjustments made. \[ \boxed{ f(x) = \frac{k \cdot (x-c_1)^{p_1} (x-c_2)^{p_2} \dots}{(x-a_1)^{q_1} (x-a_2)^{q_2} \dots} } \] - Where \( k \) is any constant used to fix the y-intercept or asymptote. --- ## Example (with Sample Data) Suppose the graph has: - Vertical asymptotes at \( x = -2 \) and \( x = 3 \) - X-intercept at \( x = 1 \) (crosses), \( x = 4 \) (touches) - y-intercept at \( y = 2 \) - Horizontal asymptote at \( y = 0 \) **A.** - Vertical asymptotes: \( x = -2 \), \( x = 3 \) - Factors: \( (x+2) \), \( (x-3) \) in denominator **B.** - X-intercepts: \( x = 1 \) (cross) \( \rightarrow (x-1)^1 \), \( x = 4 \) (touch) \( \rightarrow (x-4)^2 \) in numerator **C.** - Trial function: \[ f(x) = \frac{(x-1)(x-4)^2}{(x+2)(x-3)} \] - y-intercept: \( f(0) = \frac{(0-1)(0-4)^2}{(0+2)(0-3)} = \frac{(-1)(16)}{(2)(-3)} = \frac{-16}{-6} = \frac{8}{3} \) - To get y-intercept 2, multiply by \( k \): \[ 2 = k \cdot \frac{8}{3} \implies k = \frac{3}{4} \] So, \[ f(x) = \frac{3}{4} \cdot \frac{(x-1)(x-4)^2}{(x+2)(x-3)} \] - Horizontal asymptote: degrees numerator = 1+2=3, denominator = 1+1=2, so no horizontal asymptote at \( y=0 \). If you need \( y=0 \), adjust powers. **D.** - **Final function (after all adjustments):** \[ \boxed{ f(x) = \frac{3}{4} \cdot \frac{(x-1)(x-4)^2}{(x+2)(x-3)} } \] --- If you provide your actual graph’s features, I can fill in the steps with your numbers.