Professor Baker's Math Class - August 31, 2023: Chapter 1 Review
Welcome back to math class! Today, we dove deep into several key concepts from Chapter 1. Let's recap what we covered to solidify your understanding.
Topics Covered
- Finding x- and y-intercepts of a line given the equation: Advanced
Remember that the x-intercept is the point where the line crosses the x-axis (where $y = 0$), and the y-intercept is where the line crosses the y-axis (where $x = 0$).
For example, given the equation $5x + 6y = -7$, to find the y-intercept, set $x = 0$: $6y = -7$, so $y = -\frac{7}{6}$. The y-intercept is $(0, -\frac{7}{6})$. To find the x-intercept, set $y = 0$: $5x = -7$, so $x = -\frac{7}{5}$. The x-intercept is $(-\frac{7}{5}, 0)$.
- Writing the equation of a line through two given points
First, find the slope ($m$) using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$. Then, use the point-slope form of a line: $y - y_1 = m(x - x_1)$.
For the points (5, -6) and (-5, -1), the slope $m = \frac{-1 - (-6)}{-5 - 5} = \frac{5}{-10} = -\frac{1}{2}$. Using the point-slope form with the point (5, -6): $y + 6 = -\frac{1}{2}(x - 5)$.
- Writing and evaluating a function that models a real-world situation: Advanced
Identify the variables and their relationship. Look for keywords that indicate slope (rate of change) and y-intercept (initial value).
Example: Ashley's total pay ($P$) is a function of the number of "Math is Fun" copies she sells ($N$). If her base salary is $2500 and she makes $120 per copy, the equation is $P = 120N + 2500$. If she sells 29 copies, her total pay is $P = 120(29) + 2500 = 3480 + 2500 = $5980.
- Writing an equation and drawing its graph to model a real-world situation: Advanced
Similar to above, but also involves visualizing the relationship on a graph. Define your axes and plot points accordingly.
Example: A cookie company uses one cup of sugar for every 25 cookies. Let $S$ represent the total number of cups of sugar and $N$ the number of cookies. Then $S = \frac{1}{25}N$.
- Finding the intercepts and rate of change given a graph of a linear function
Identify the points where the line crosses the x and y axes for intercepts. The rate of change is the slope, calculated as rise over run between two points on the line.
- Choosing a graph to fit a narrative: Advanced
Consider the key features of the narrative: increasing/decreasing values, constant values, maximum/minimum values, and how these translate to the shape of a graph.
- Identifying parallel and perpendicular lines from equations
Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other (e.g., if one slope is $m$, the other is $-\frac{1}{m}$).
- Identifying parallel and perpendicular lines from coordinates
Calculate the slopes between the points. Apply the same rules as above (same slope for parallel, negative reciprocal slopes for perpendicular).
- Identifying coordinates that give right triangles
Use the distance formula (or Pythagorean theorem) to find the lengths of the sides. Check if $a^2 + b^2 = c^2$ to confirm if it's a right triangle.
- Evaluating functions: Absolute value, rational, radical
Substitute the given value into the function and simplify. Remember to follow the order of operations.
- Domain and range from the graph of a continuous function
The domain is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values). Look for endpoints and any restrictions on the graph.
If the graph of a function has endpoints at (-3, 0) and (3, 4), the domain would be [-3, 3], and the range would be [-5, 4]
- Domain of a rational function: Interval notation
A rational function is a fraction where the denominator cannot be zero. Set the denominator not equal to zero and solve for x. Express the domain in interval notation.
For example, for the function $f(x) = \frac{x+1}{x^2 - 1}$, we must ensure $x^2 - 1 \neq 0$. This means $(x+1)(x-1) \neq 0$, so $x \neq -1$ and $x \neq 1$. The domain is $(-\infty, -1) \cup (-1, 1) \cup (1, \infty)$.
- Domain of a square root function: Advanced
The expression inside the square root must be greater than or equal to zero. Set the expression $\ge 0$ and solve for x. Express the domain in interval notation.
- Finding where a function is increasing, decreasing, or constant given the graph
Observe the graph from left to right. If the y-values are increasing, the function is increasing. If the y-values are decreasing, the function is decreasing. If the y-values are constant, the function is constant.
Keep practicing, and you'll master these concepts in no time! Remember, math is a journey, not a destination. See you in the next class!