Math Problem Statement
Solution
To determine which function has the same y-intercept as the graph shown, we need to find the y-intercept of the line in the image, then check each equation to see which one shares this y-intercept.
Step-by-Step Solution
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Identify the y-intercept from the graph: The graph shows a line crossing the y-axis. From visual inspection, it looks like the y-intercept is around .
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Rewrite the equations in slope-intercept form (y = mx + b) if necessary to identify their y-intercepts:
- (1) : Simplify this to find , which gives a y-intercept of .
- (2) : Rearrange to solve for , resulting in , so the y-intercept is .
- (3) : Rearrange to get , giving a y-intercept of .
- (4) : Simplify this to , which gives a y-intercept of .
Conclusion
The correct answer is Option (4) , as it has the same y-intercept of as the graph.
Would you like more details on any of these steps?
Here are some follow-up questions that might deepen your understanding:
- How do you identify the y-intercept of a line from an equation?
- What are the steps to rearrange an equation to slope-intercept form?
- Why is it useful to express a line in slope-intercept form?
- What does the slope of a line represent in an equation?
- How can you determine the slope and y-intercept directly from a graph?
Tip: Always look for the y-intercept by setting in an equation when identifying it in non-standard forms.
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Y-Intercept
Slope-Intercept Form
Formulas
y = mx + b
Theorems
Slope-Intercept Form of a Linear Equation
Suitable Grade Level
Grades 8-10