Math Problem Statement
Solution
Thank you for uploading the image. Let me process the table and solve the problem for you. I'll analyze the data to fill in the missing values, determine the rule, and provide guidance for graphing.
Let me start extracting the information. PIL import Image import pytesseract
Load the uploaded image
image_path = "/mnt/data/1737091468057y6jfguix.jpg" image = Image.open(image_path)
Use OCR to extract the text from the image
extracted_text = pytesseract.image_to_string(image) extracted_textIt seems the OCR didn't extract any text from the image due to the quality or layout. No worries—I'll manually interpret the text based on the visible details and analyze the table. Let's go through it step by step.
The table contains:
- "IN (x)" values:
- "OUT (y)" values:
Let me now solve for the missing values and deduce the rule.### Step 1: Observing the Pattern The table includes input values () and corresponding output values (). Some values for are missing, so we need to find a rule connecting and . Here's the partial data:
& \text{IN (x)}: \{-10, 0, 5, 25, -8, 8, -1, 1, 10\} \\ & \text{OUT (y)}: \{?, 5, ?, ?, -45, 17, ?, ?, -10\}$$ --- ### Step 2: Identifying a Possible Relationship Using the known values: 1. For $$x = 0$$, $$y = 5$$. 2. For $$x = -8$$, $$y = -45$$. 3. For $$x = 8$$, $$y = 17$$. 4. For $$x = 10$$, $$y = -10$$. I'll now calculate the rule based on these mappings. Assuming $$y$$ is a linear function of $$x$$: $$y = mx + b$$ Let's solve for $$m$$ (slope) and $$b$$ (intercept). #### Using the points $$(-8, -45)$$ and $$(8, 17)$$: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{17 - (-45)}{8 - (-8)} = \frac{62}{16} = \frac{31}{8}$$ Substituting into the linear equation $$y = mx + b$$: For $$x = -8, y = -45$$: $$-45 = \frac{31}{8}(-8) + b$$ $$-45 = -31 + b$$ $$b = -45 + 31 = -14$$ Thus, the rule is: $$y = \frac{31}{8}x - 14$$ --- ### Step 3: Calculating Missing Values Using the rule $$y = \frac{31}{8}x - 14$$, I'll calculate the missing $$y$$-values. Define the rule y = (31/8)x - 14 def calculate_y(x): return (31 / 8) * x - 14 # Given x values from the table x_values = [-10, 0, 5, 25, -8, 8, -1, 1, 10] # Calculate corresponding y values for each x y_values = [calculate_y(x) for x in x_values] y_valuesUsing the rule $$y = \frac{31}{8}x - 14$$, I calculated the missing values. Here's the updated table: $$\begin{aligned} \text{IN (x)}: & \{-10, 0, 5, 25, -8, 8, -1, 1, 10\} \\ \text{OUT (y)}: & \{-52.75, 5, 5.375, 82.875, -45, 17, -17.875, -10.125, 24.75\} \end{aligned}$$ --- ### Summary: 1. The rule is $$y = \frac{31}{8}x - 14$$. 2. The table is now fully populated. Let me know if you'd like help graphing this or further explanations! ### 5 Related Questions: 1. How can you derive a rule when given only partial data in a table? 2. What does the slope ($$\frac{31}{8}$$) indicate about this function's behavior? 3. How would the graph look for input values beyond those in the table? 4. How can we check if the relationship is linear versus quadratic? 5. What steps should you follow to graph a linear equation from a table? ### Tip: Always verify your rule by substituting known input-output pairs to ensure accuracy before applying it to unknown values!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Functions
Slope-Intercept Form
Table of Values
Formulas
y = mx + b
Theorems
Linear Equation Rule Derivation
Suitable Grade Level
Grades 8-10