1. **Problem stated:** Make a one-page cheat sheet from the pages about **slope, slope-intercept form, graphing linear equations, functions, and relations**. 2. **Key formulas to know:** $$m=\frac{y_2-y_1}{x_2-x_1}$$ $$y=mx+b$$ $$x=a\text{ means a vertical line}$$ $$y=b\text{ means a horizontal line}$$ 3. **Slope rules:** - Slope is **rise over run**. - Positive slope means the line goes **up** from left to right. - Negative slope means the line goes **down** from left to right. - Horizontal lines have slope $0$. - Vertical lines have **undefined** slope. 4. **How to find slope from two points:** - Pick two points. - Substitute into $m=\frac{y_2-y_1}{x_2-x_1}$. - Simplify. - If the denominator becomes $0$, the slope is undefined. 5. **How to write an equation in slope-intercept form:** - Start with $y=mx+b$. - Find the slope $m$. - Use one point to solve for $b$. - Write the final equation. - Check your answer by substituting a point from the table or graph. 6. **How to graph $y=mx+b$:** - Plot the y-intercept $(0,b)$. - Use the slope as $\frac{\text{rise}}{\text{run}}$. - Move up/down for rise and right/left for run. - Plot another point and draw the line. 7. **Intercepts for graphing in standard form:** - To find the x-intercept, set $y=0$. - To find the y-intercept, set $x=0$. - Plot both intercepts and draw the line. 8. **What a table can tell you:** - If the change in y is constant for equal changes in x, the relation is linear. - Constant rate of change means a straight line. - Varying rate of change means the relation is nonlinear. 9. **Function rules:** - A function gives **exactly one output** for each input. - If one x-value is paired with more than one y-value, it is **not** a function. - Use the **vertical line test** on graphs. - If a vertical line crosses the graph more than once, it is not a function. 10. **Linear vs. nonlinear:** - Linear graphs are straight lines. - Nonlinear graphs are curved or not straight. - Examples of nonlinear graphs include parabolas and curves like $y=x^2$ or $y=\sqrt{x}$. 11. **Real-world meaning of slope and intercept:** - The y-intercept is the **starting value**. - The slope is the **rate of change**. - In word problems, identify what x and y represent before writing the equation. 12. **Quick examples from the pages:** - Sarah’s wages: $y=2x+5$. - Keri’s wages: $y=x+10$. - Sand pile: $y=-5x+600$. - Pool draining: $y=3200-25x$. - Cab fare example: cost increases by a fixed amount per mile plus an initial fee. 13. **Fast checklist for solving linear equation problems:** - Find the slope. - Find the y-intercept. - Write $y=mx+b$. - Check with a point. - Interpret what the slope and intercept mean in context. 14. **Cheat-sheet summary:** - Slope = rate of change. - Y-intercept = starting value. - Use $m=\frac{y_2-y_1}{x_2-x_1}$ to find slope. - Use $y=mx+b$ to write equations. - Use intercepts or slope-intercept form to graph lines. - A function has only one output for each input.