Quadratic Functions – What You Need to Know

Most intermediate-advanced algebra courses cover second degree polynomial functions, called “quadratic functions,” long before introducing general polynomial functions.

 

Because quadratic functions are simpler versions of general polynomial functions, graph as parabolic curves, and are featured in many real-world situations, this subtopic holds a prominent place in the syllabus.

Following is a list of concepts you should have firmly under your belt before taking the SAT/ACT.

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[Note: “iff” means “if and only if.”]

Definition
Quadratic functions: functions involving one first-degree variable and one second degree variable (typically y^1 and x^2).

Quadratic Function Standard Form
y = ax^2+bx+c.

Parabola
Vertex = (h,k) with h = -b/2a, k = f(h).
Axis: x = h.
y-intercept = c.

Vertex Form
y = a(x–h)^2+k.
Vertex = (h,k).
Radius = r.

Discriminant
D = sqrt(b^2–4ac).
D > 0 iff 2 real solutions.
D = 0 iff 1 real solution.
D < 0 iff no solutions.

Sum and Product Formula
x^2–(sum)x+(prod) = 0.

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For practice, search Google for worksheets covering any or all of the above, pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered this important material.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Factfulness

Despite rumors to the contrary, things aren't so bad. They're probably quite a bit better than you think. That's the thesis of this book by Swedish author and public health expert Hans Rosling.

From the book's description on amazon:

"It turns out that the world, for all its imperfections, is in a much better state than we might think. That doesn’t mean there aren’t real concerns. But when we worry about everything all the time instead of embracing a worldview based on facts, we can lose our ability to focus on the things that threaten us most.

"Inspiring and revelatory, filled with lively anecdotes and moving stories, Factfulness is an urgent and essential book that will change the way you see the world and empower you to respond to the crises and opportunities of the future."

Here's what Bill Gates had to say about Factfulness:

“One of the most important books I’ve ever read―an indispensable guide to thinking clearly about the world.”

And Barack Obama:

"Factfulness by Hans Rosling, an outstanding international public health expert, is a hopeful book about the potential for human progress when we work off facts rather than our inherent biases."

Rosling isn't a Pollyanna who refuses to acknowledge the very real problems and challenges humanity still faces. His point is that, if we actually follow the facts – instead of persistent, unconscious biases – there's much more reason to be proud and hopeful than we might ordinarily believe.

I enjoyed Factfulness, and highly recommend it to all with an interest in the state of the world.

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Contact Chris

borlandeducational.com

Copyright © 2006-present: Christopher R. Borland. All rights reserved.