Linear and Quadratic Functions as Models

This is Unit Six of College Algebra: linear and quadratic functions as models. Unit Five built the general machine. This unit takes the two simplest kinds of it and does something the course has not done before — points them at situations outside mathematics and asks what they can tell you. The capstone puts the division cleanly: the line handles change that holds steady, and the parabola handles change that turns around. Which family fits is decided by the situation rather than by preference, and the entries are careful about that. Two cautions run through the unit and both are about honesty rather than technique. A rate quoted without its units is half a fact. And a model used beyond the data it was built from is being trusted rather than tested. By the end you will be able to do five things. One: build a linear model from a rate and a starting value, and say when one does not apply. Two: say why predicting inside your data differs from predicting beyond it. Three: read a parabola's vertex without inverting the sign. Four: say why one object carries three different names. Five: optimise with a quadratic, and answer the question actually asked.

Three entries reintroducing the line, now as a function rather than a graph. The two numbers in it were geometry in Unit Three; here they become a rate and a starting value, which is what makes the family useful for anything real.

The family.

Alg 301 Linear Functions

A function whose graph is a line, described by two numbers — the steepness, and the output when the input is zero.

What the first number says.

Alg 302 Direction from the Slope

A positive slope means outputs climb throughout, negative means they fall throughout, and zero means the function never changes at all.

What they mean in a situation.

Alg 304 Slope as a Rate

The slope becomes a rate — so much output for each unit of input — and the constant becomes the value at the start. Quote a rate without saying per what, and you have given half a fact.

Two procedures, one from a described situation and one from data — and the second carries the most honest sentence in the unit, about the difference between a prediction you have evidence for and one you do not.

Building from a situation.

Alg 306 Building a Linear Model

Find the steady rate and the starting value, assemble the function, then test it against a case you know. The steady rate is the entry requirement — where the rate itself changes, this family does not fit and a different one is needed.

Building from data.

Alg 308 A Linear Model from Two Data Points

Turn the observations into coordinate pairs, run the two-point construction, and interpret the result. Between your data points the model is supported. Beyond them it is being trusted on credit, and the entry says so plainly.

Two questions asked of linear models constantly. When do two of them agree, and when does one reach zero — and the second has an exact count attached, which not every function family can offer.

Where two agree.

Alg 310 Where Two Lines Meet

The two outputs match where the formulas do, so setting them equal and solving locates the input where the models cross.

Where one reaches zero.

Alg 312 The Zero of a Linear Function

The input producing an output of zero. Any line that is not flat has exactly one — no more, and none missing.

The second family, for change that reverses direction. Its whole description hangs on one point, and everything the rest of the unit does is either finding that point or reading something off it.

The family.

Alg 321 Quadratic Functions and Parabolas

A degree-two function, graphing as a parabola that opens upward when the leading number is positive and downward when it is negative.

The point everything hangs on.

Alg 322 The Vertex and Axis of Symmetry

The turning point — the lowest point when it opens upward and the highest when it opens down — with a vertical line of symmetry running through it.

Two ways of writing the same function. One is convenient for arithmetic and conceals the vertex; the other displays the vertex directly. And displaying it comes with a sign behaving the way the horizontal shifts did in Unit Five — which is to say, misleadingly.

The two forms.

Alg 324 General Form and Standard Form

The expanded arrangement, and the one built around the vertex. Some books call the second one vertex form, which is the more descriptive name.

Reading it off.

Alg 325 Reading the Vertex from Standard Form

The number inside the bracket gives the first coordinate with its sign reversed, and the number outside gives the second coordinate as written. The inside misleads exactly as the shifts did.

The error.

Alg 326 The Vertex-Sign Error

Reading the inside number without flipping it. Where the bracket shows an addition, rewrite it as a subtraction of a negative first — then the reading is mechanical and the sign takes care of itself.

Finding it from the other form.

Alg 327 The Vertex Formula

A formula gives the first coordinate directly from the coefficients, and the axis of symmetry sits at that same value. Get the second coordinate by evaluating the function there — never by reading off the constant term.

One procedure, and it is Unit Four's completing the square with a single difference that catches people every time — there is no equals sign here, so there is nothing to balance against.

Watch the conversion.

Alg 329 Converting General to Standard Form

Group the variable terms, add the completing amount, and subtract it again immediately so the function is unchanged. In Unit Four the same amount went onto both sides — a function has no second side, so it goes on and comes off in one place.

Two entries on what the vertex is worth. It supplies the extreme value directly, which is what makes the family useful for optimisation. And then an entry about vocabulary that resolves a confusion the course has been quietly building.

The extreme value.

Alg 331 Maximum and Minimum Values

The vertex's output is the smallest value when the parabola opens upward and the largest when it opens down — read from the point, with no searching required.

Three names, one thing.

Alg 333 Zeros, x-Intercepts, Solutions

The same value is called one thing when you are thinking about the function, another when thinking about the graph, and a third when thinking about an equation. The names record the lens, not a difference — and the discriminant counts how many there are.

Two procedures to close the working. The first reads every feature of a parabola off its formula in a fixed order. The second uses the vertex on a real question, and its warning is about the last step rather than the mathematics.

Reading everything off.

Alg 335 The Full Quadratic Workup

Opening direction from the leading coefficient, vertex and axis from the formula, the vertical intercept from the constant, and the horizontal ones from solving. The symmetry check costs nothing — intercepts sitting unevenly around the axis are announcing an error.

Using it on a question.

Alg 337 Optimizing with a Quadratic

Write the quantity as a quadratic in one variable, find the vertex, and read the extreme value. Then check what was asked — the vertex supplies the input that optimises, and the question may have wanted the output, or something built from both.

The unit closes by naming the division it has been working with — two families, two kinds of change — and stating that every feature of both was read rather than guessed at.

The capstone. Two families, at work.

Alg 350 The Parabola Dossier

The line models change that holds its rate; the parabola models change that reverses. Every feature of both comes out of the formula by a stated procedure — and every claim here carries an earlier number.

Five things you can now do. You can build a linear model from a rate and a starting value, and you know the entry requirement — a rate that holds steady, since a changing one needs a different family entirely. You can say why predicting inside your data differs from predicting beyond it: between the points the model is supported, and past them it is trusted. You can read a vertex from standard form, rewriting an addition inside the bracket as a subtraction first so the sign takes care of itself. You can say why one object has three names — the names record which lens you are using, not a difference in the thing. And you can optimise with a quadratic, then check the question, because the vertex hands you the optimising input and the question may want something else.