# Week 12 – Precalc 11

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.

Example:

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.

Example:

# Week 11 – Precalc 11

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.

Example:

# Week 10 – Precalc 11

This week in Precalculus 11, we started the Graphing Inequalities & Systems of Equations unit. We first learned about solving quadratic inequalities in one variable.

quadratic inequality: an expression with degree 2 that is not equal to 0

It can be written in general form as a$x^2$ + bx + c >/</≥/≤ 0

To solve a quadratic inequality, we first rearrange the inequality to general form. We then write and solve its corresponding quadratic equation. We label a number line with the roots. We then test a number from each interval by substituting it into the original inequality.

Example:

This was new to me, as I had learned how to solve linear inequalities, but not quadratic inequalities.

# Week 9 – Precalc 11

This week in Precalculus 11, we learned about equivalent forms of a quadratic function.

x-intercept: point on a graph crossing the x-axis

root: x-value to an equation when solved

y-intercept: point on a graph crossing the y-axis

vertex: maximum (opening down) or minimum (opening up) point on a parabola

A quadratic function has 3 forms: factored, general, and standard.

factored form: y = a(x – $x_1$)(x – $x_2$)

• $x_1$ & $x_2$ are the graph’s x-intercepts, or the equation’s roots

general form: y = a$x^2$ + bx + c

• c is the graph’s y-intercept

standard form: y = a$(x - p)^2$ + q

• p is the vertex’s x-value
• q is the vertex’s y-value

For all 3 forms, a determines whether the parabola opens up (positive) or down (negative) and its stretch (a>1) or compression (a<1) value.

By converting from one form to another, we can determine all elements of the equation.