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 firmly understand 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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borlandeducational.com

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.

Winning Focus

Test taking is a competitive activity. 

As is athletics or the performing arts, outstanding achievement on the SAT/ACT depends on maintenance of powerfully sharp, energetic, mindful concentration and “winning focus.” 

In sports, the challenge is the opposing team. In the performing arts, it’s stage fright. In test taking, your adversaries are the test questions. 

Test takers, like athletes and stage performers, need to confront and conquer pressure, stress, fatigue, distractions, and other difficulties with sustained enthusiasm, relentless drive, and total commitment to success. If victory is the object, nothing less will do.

An intense, aggressive, yet controlled state of concentration is critical to performing at one’s best in any competitive activity. “Winning Focus” is so important, in fact, that without it, almost nothing else matters.

By definition, intensity always involves some amount of tension. Undisciplined, raw intensity can lead to varying degrees of frustration, anxiety, and even exhaustion. Athletes, performers, and test takers must learn to thread the needle between two extremes, remaining fully responsive and focused on winning without getting rattled or falling victim to tenseness, tiredness, or negative emotion.

When intelligence is inhibited by inattention, distraction, tiredness, or emotional upset, reduced cognitive bandwidth causes a mental traffic jam, disrupting the efficient flow of ideas.

On the other hand, Winning Focus lights up the brain like a Christmas tree. All kinds of connections, insights, and creative solutions come to mind that would otherwise go unnoticed. Most important of all, careless mistakes are minimized.

Fully half the errors on a typical student's SAT/ACT math test are due to carelessness. If these careless errors were reduced by half, a student scoring 650 on an SAT math section would instead score 720, and one scoring 27 on the ACT would score 31!

These dramatic improvements require neither new math smarts, nor improved testing acumen, nor additional practice – just a 50% cutback in careless mistakes. In fact, math test prep could be restricted entirely to development of Winning Focus, and most students would be well served.

The vast majority of test coaches pay no significant attention to the mental game of test taking. In my work, Winning Focus takes center stage. Using proprietary methods and materials, a natural knack for inspiring and igniting motivation, and a unique ability to train all aspects of effective test prep that only 45+ years in business can bring, I teach my students to track errors, avoid unnecessary mistakes, and generate and sustain powerful, energetic, productive concentration.

The best way to train Winning Focus is to build strength and endurance through gradual exposure to practice tests of increasing length and difficulty, bolstered by professional performance critiquing, feedback, error analysis, and personal support. It boils down to habituating a determined, enthusiastic, mindful concentration on test questions, and training higher levels of focus for longer and longer periods.

Fortunately, most high-performing students already know how to do this. Such students are typically athletes, actors, dancers, and/or musicians, and already know how to generate the state of mind necessary to excel at competitive activities. It’s just a matter of transferring this well-developed ability to the domain of test taking. Although the particulars are more complex, it’s really that simple.

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Contact Chris

borlandeducational.com

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

lizardpoint

Lizardpoint is a wonderful all-around early learning site for people of all ages. Students can use lizardpoint to learn geography, flags, world leaders (current and past), art, and math.

The site is great for adults as well as for children.

Sections on geography help students master finding countries and provinces, naming their capitals, topographical features, airport codes, and more.

The section on art features quizzes on major western artists and their most well-known works, fine art terminology, and important movements.

Math offerings include on-screen and downloadable/printable worksheets (with answer sheets) for arithmetic operations, decimals, percents, and fractions.

Quizzes on trivia, important definitions, and more round out learning activities presented.

New stuff is added regularly, collected in What's New.

Like Freerice.com, lizardpoint can be addicting, but in a good way. These two educational sites keep interested students learning and growing while having fun.

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Contact Chris

borlandeducational.com

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

Linear Functions – What You Need to Know

Linear functions is one of the most frequently tested subtopics on the SAT/ACT, and students continually employ these basic functions as they progress through more and more rigorous math courses. Indeed, there’s nothing more important to master in high school mathematics than linear models and methods.

But which ideas should students fluently understand and keep in mind to score well on the SAT/ACT?

Here’s the list.

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Definition
Linear Functions: functions involving two first-degree variables (typically y^1 and x^1).

Vertical line equations
x = #. 

For example: x = 4 graphs as a vertical line through 4 on the x axis.

Horizontal line equations
y = #. 

For example: y = -4/5 graphs as a horizontal line through -4/5 on the y axis.

Oblique lines

Linear equation standard form: Ax+By = C, slope = -A/B. 

For example: 3x – 4y = 12, slope = -(3/-4) = 3/4.

Slope-intercept form: y = mx + b, slope = m = "rise/run," y-intercept = b. 

For example: y = -x, slope = -1 = -1/1, y-intercept = 0.

Parallel Lines
m1 = m2 (equal slopes).

Perpendicular Lines
m1 * m2 = -1 (opposite reciprocal slopes).

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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.

-----

Contact Chris

borlandeducational.com

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