Week 9 – Precalc 11 – Caleb’s Blog

The first half of Week 9 was focused on the preparation of the midterm. After the midterm, we learned some new concepts in Chapter 4 “Analyzing Quadratic Functions and Inequalities.”

First, we reviewed the definitions of parabola, vertex, axis of symmetry, domain, and range. Then, we learned the Standard Form of the equation of a quadratic function:

$y = a(x -p)^2 + q$

Each term in this equation can transform the graph in its unique way. Let’s take a basic equation or parent function $y=x^2$ (which is a parabola graph) and create a completely different graph by modifying the terms.

The parabola below illustrates the parent function $y=x^2$ where the coefficient of the first term ( $x^2$ ) is positive.

If the coefficient of the first term ( $x^2$ ) is a negative, the vertex will open down and will become a maximum point.

The graph below is stretched vertically when the coefficient ( $a$ ) of the first term ( $x^2$ ) is greater than 1. The example compares the stretch with the parent function and how it has changed. The red parabola appears narrower.

However, the graph can be widened vertically when the coefficient ( $a$ ) of the first term ( $x^2$ ) is between 0 and 1. The example below compares the stretch with the parent function and how it has changed.

If the value of $p$ is positive, the graph moves to the left (horizontally) by that quantity. The example below compares the horizontal translation with the parent function and how it has changed.

And, if the value of $p$ is negative, the graph moves to the right (horizontally) by that quantity. The example compares the horizontal translation with the parent function and how it has changed.

If the value of $q$ is positive, the whole graph moves up by that quantity. The example compares the vertical translation with the parent function and how it has changed.

If the value of $q$ is negative, the whole graph moves down by that quantity. The example compares the vertical translation with the parent function and how it has changed

The following graph is a perfect example of putting everything we learned from a function into a graph.

The final lesson of this week is similar to last week’s hack on “completing the square” for an equation. This time, we will use this hack to re-write an equation in Standard Form.
We will use question # 6a. from Unit 4.5 as an example:

$y = 2x^2 + 8x - 4$

First, we must factor out the greatest common factor among the first and second terms of the equation only! The third term or the constant remains untouched.

$y=2( x^2 + 4x) -4$

To find a “perfect square” trinomial on the left side of the equation, we must divide the coefficient of the second term ( $4x$ ) by 2 and then square that value.

Write this value inside the bracket so that the perfect square can be completed. Also, write the same value again with a subtraction sign.

Now, we can ‘complete the square’, multiple the last term by 2 inside the square brackets, and finally add the remaining constants together to obtain the Standard Form of the expression.

In this example, we now know that the coordinates for the vertex is on (-2,-12), which is reflected in the terms responsible for the vertical and horizontal translations.

Week 9 – Precalc 11

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