Unit 2, Quadratic Functions: Day 17

Focus: Graphing Quadratics from Intercept Form

Action Items:

1. Show me yesterday's homework 2.15.

2. Take good notes on graphing.

3. Homework 2.16: Summary Worksheet, Intercept Form.

Lesson Goal: At the end of this lesson, you will be able to graph quadratic equations by hand from the Intercept Form.

Today's Lesson: Intercept Form of a Quadratic Equation, Graphing, Summary, and Review

$LaTeX: y=a\left(x-p\right)\left(x-q\right)$
Domain: $LaTeX: x\in\mathbb{R}$
Range: $LaTeX: y\ge$ the y-value of the vertex when OR $LaTeX: y\le$ the y-value of the vertex when .

Things that are easy to see from the Intercept Form:

The intercepts! $LaTeX: \left(p,\:0\right)$ and $LaTeX: \left(q,\:0\right)$
Just like the other two forms: The value of gives you information about whether the parabola opens up (a is positive), opens down (when a is negative), and its rate of growth.
- If (in other words, if it is positive), the parabola opens up ("opens up = makes a cup"), and the curve has a lowest point, or minimum value.
- If (in other words, if it is negative), the parabola opens down ("opens down = makes a frown"), and the curve has a highest point, or maximum value.
- If , the parabola grows faster than normal, so it appears more narrow.
- If (in other words, a fraction less than 1), the parabola grows more slowly, so it appears wider.

Harder to see, but necessary:

The equation for the Axis of Symmetry is $LaTeX: x\:=\frac{p+q}{2}$ . Notice: the Axis of Symmetry must be halfway between the intercepts, so we use the midpoint formula to find the x-value of the vertex. In other words, add the intercept values and divide by 2.
To find the VERTEX, plug the value $LaTeX: x\:=\:\frac{p+q}{2}$ into the equation and solve for y.

Connections to Solving:

The intercept form is one step away from being already solved: set each factor equal to zero and solve for x to find the intercepts.
Those intercepts are equidistant to the Axis of Symmetry (the Axis of Symmetry is half way between the x-intercepts).

For every quadratic equation you graph, you should

plot and identify the vertex
plot two points on either side of the vertex (use symmetry), ideally
- two of those points should be the x-intercepts (if any) -- these will be approximations if the roots are irrational.
- if identifying the intercepts is difficult (because they are irrational), use the rate of growth.
in order to be complete, your parabola should show the vertex and intercepts even if you do not label those

Homework 2.16: Write the answers in your composition book: Summary Worksheet, Intercept Form

Resources:

Quadratic Unit Homework Solutions

Reminder: all of the resources can be found in the Quadratics Resources Module.

Also, try Patrick JMT Math Tutorials: All About Quadratics (Complete Playlist)

And if you like Kahn Academy, use their Quadratics unit if you need more support.