This week in Precalculus 11, we learned about the Sine Law. It gives us a more efficient method to solving triangles without 90° angles. However, it can only be used if an angle, its opposite side length, and one other piece of information are given.
The Sine Law is commonly written in 2 forms. The first is used when solving for a side length, and the second is used when solving for an angle.
When only given one angle using Sine Law, there could be 2 different triangles. Solve for an angle, and if it is less than 90°, find its quadrant II coterminal angle. Add this to the originally given angle, and if it is less than 180°, there are 2 possible triangles.
This week in Precalculus 11, we learned about solving rational equations.
Rational equation: an equation containing rational expressions.
To solve a rational equation, we first identify non-permissible values. One way to solve a rational equation is to multiply each term by the lowest common denominator, then solve normally. Another is if 2 fractions are equal to eachother and have either the same numerators or same denominators, their numerators/denominators must also match. We then check for extraneous roots, or non-permissible values.
This week in Precalculus 11, we started the Rational Expressions and Equations Unit. We first learned about equivalent rational expressions.
rational expression: a fraction with polynomials in the numerator & denominator
non-permissable value: variable values making the denominator 0
To determine the non-permissable values, we equate the denominator to 0 and solve the equation.
To simplify a rational expression, we first factor the numerator and denominator. We then identify the non-permissable values. Last, we cancel out identical pairs of numerators and denominators.
This week in Precalculus 11, we learned about graphing reciprocals of linear functions.
reciprocal: numerator and denominator are switched
linear function: an equation in 2 variables with degree 1
asymptote: barrier lines of a graph
To graph the reciprocal of a linear function, we first graph its parent function. We then find the points on the graph where y = 1 and y = -1, and the vertical and horizontal asymptotes. In most cases, the horizontal asymptote is y = 0. The vertical asymptote is the x-intercept. We then use these points to draw a 2-part graph known as a hyperbola. The asymptotes act as barriers.
This week in Precalculus 11, we started the Absolute Value and Reciprocal Functions unit. We first learned about absolute value functions.
Absolute value function: y = |f(x)|
Critical points: an absolute value function’s x-intercepts
An absolute value function’s graph changes direction at the critical points. To graph an absolute value function, we first graph its equivalent normal function [y = f(x)]. Any points below the x-axis are reflected along the x-axis, since absolute values are only positive.
An absolute value function can be written in piecewise notation. To do so, we write one function in which the absolute value expression is positive or 0, and one function in which the absolute value expression is negative. We then state when this occurs, using </>/≥/≤ and the critical point.
This week in Precalculus 11, we learned about graphing linear inequalities in 2 variables.
linear inequality: written in general form as ax + by + c </>/≤/≥ 0
To graph a linear inequality in 2 variables, we first graph its corresponding linear equation. To do this, we must rearrange the equation to standard form, where y = mx + b.
We then plot the y-intercept (b) and use the slope (m) to plot more points. We connect these points with a straight line. If the symbol is < or >, it means less than or greater than, and we use a broken line. If the symbol is ≤ or ≥, it means less than or equal to or greater than or equal to, and we use a solid line.
One side of the line has solutions, and the other does not. To determine which side does, we plug a test point (labelled T) into the original equation. (0, 0) is a good test point. If the statement is true, we shade in the side of the graph with said point. If not, we shade in the other side.
This was interesting, as I had learned how to graph linear equations, but not inequalities.