Algebra and graphs

Welcome to Algebra and Graphs!

In this chapter, we are going to explore how to turn complex equations into visual stories. We’ll look at rational functions (fractions with variables) and conic sections (cool shapes like ellipses and hyperbolas). Understanding these graphs is like having a map; it helps you see exactly how numbers behave. Don't worry if it looks intimidating at first—once you learn the "landmarks" (like asymptotes and intercepts), sketching these becomes much easier!

1. Rational Functions

A rational function is simply a fraction where the top (numerator) and the bottom (denominator) are polynomials. In FP1, we focus on three specific types:

1. \(y = \frac{ax+b}{cx+d}\) (Linear over Linear)
2. \(y = \frac{ax+b}{cx^2+dx+e}\) (Linear over Quadratic)
3. \(y = \frac{x^2+ax+b}{x^2+cx+d}\) (Quadratic over Quadratic)

Asymptotes: The "Invisible Fences"

Asymptotes are lines that the graph approaches but never actually touches (usually!). Think of them as magnetic fences that guide the shape of your curve.

• Vertical Asymptotes: These happen where the denominator equals zero. Since we can't divide by zero, the graph "explodes" towards infinity at these points.
  Example: For \(y = \frac{1}{x-2}\), the vertical asymptote is \(x = 2\).
• Horizontal Asymptotes: These show what happens to \(y\) when \(x\) becomes very, very large (positive or negative).

How to Sketch a Rational Function (Step-by-Step)

1. Find the Intercepts: Set \(x=0\) to find the \(y\)-intercept. Set the top (numerator) to \(0\) to find the \(x\)-intercepts.
2. Find the Asymptotes: Set the bottom (denominator) to \(0\) for vertical asymptotes. For horizontal asymptotes, look at the highest powers of \(x\).
3. Check for "Crossing" points: See if the graph intersects with its own horizontal asymptote by setting the function equal to the asymptote value.
4. Test Regions: Pick a number between the asymptotes to see if the graph is "high" (positive) or "low" (negative).

Finding Range and Stationary Points (The Discriminant Trick)

Sometimes you need to find the maximum or minimum values without using calculus. We use Quadratic Theory for this!

The Process:
1. Set the function equal to a constant \(k\): \(y = k\).
2. Multiply out the denominator to create a quadratic equation in \(x\).
3. For the graph to exist, \(x\) must be a real number. This means the discriminant must be greater than or equal to zero (\(b^2 - 4ac \ge 0\)).
4. Solving this inequality for \(k\) tells you the range of the function (the possible \(y\)-values) and the \(y\)-coordinates of the stationary points.

Quick Review: To find where a graph exists, remember "Real roots mean \(\Delta \ge 0\)".

2. Conic Sections: Parabolas, Ellipses, and Hyperbolas

Conic sections are the shapes you get if you slice through a cone at different angles. They have very specific standard equations you need to recognize.

The Parabola (\(y^2 = 4ax\))

You already know \(y = x^2\). This is just a version that opens sideways. If \(a\) is positive, it opens to the right. If \(a\) is negative, it opens to the left.

The Ellipse (\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\))

An ellipse is basically a stretched circle.
- It crosses the \(x\)-axis at \((a, 0)\) and \((-a, 0)\).
- It crosses the \(y\)-axis at \((0, b)\) and \((0, -b)\).

The Hyperbola (\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\))

A hyperbola looks like two mirror-image curves opening away from each other.
- Asymptotes: Hyperbolas have diagonal asymptotes. For the standard form above, they are \(y = \pm \frac{b}{a}x\). (Check your formulae booklet for these!)

The Rectangular Hyperbola (\(xy = c^2\))

This is a special hyperbola where the asymptotes are simply the \(x\)-axis and the \(y\)-axis. It is often written as \(y = \frac{c^2}{x}\).

Did you know? The paths of planets around the sun are ellipses, while the path of a comet that only visits once is often a hyperbola!

3. Geometric Implications and Intersections

When you find the intersection of a line and a curve, you often end up with a quadratic equation. The discriminant (\(b^2 - 4ac\)) tells you the "geometry" of the situation:

• Two distinct real roots (\(\Delta > 0\)): The line cuts through the curve at two different points.
• One repeated real root (\(\Delta = 0\)): The line is a tangent to the curve (it just touches it).
• No real roots (\(\Delta < 0\)): The line and the curve never meet.

4. Transformations of Graphs

You can move or stretch these shapes using basic rules. If you have an equation for a curve, you can apply these changes:

• Translation: Replacing \(x\) with \((x-h)\) moves the graph \(h\) units to the right. Replacing \(y\) with \((y-k)\) moves it \(k\) units up.
• Stretch: Replacing \(x\) with \(\frac{x}{a}\) is a horizontal stretch by scale factor \(a\).
• Reflection: Swapping \(x\) and \(y\) reflects the graph in the line \(y = x\). This is very common when looking at inverse relations!

Common Mistake: When translating \(x \rightarrow x-3\), many students think it moves left. Remember: it's the opposite of the sign inside the bracket for horizontal moves!

Summary: Key Takeaways

1. Rational Functions: Always find the vertical asymptotes (bottom = 0) and horizontal asymptotes (long-term behavior) first.
2. Range: Use the \(y=k\) method and set the discriminant \(\Delta \ge 0\) to find max/min points without calculus.
3. Conics: Memorize the standard forms of the Ellipse (\(+\)) and Hyperbola (\(-\)).
4. Intersections: Use the discriminant of the resulting quadratic to determine if a line is a tangent, a secant (cuts twice), or doesn't touch the curve at all.

Don't worry if this seems tricky at first! The best way to master this is to practice sketching these shapes. Once you've drawn a few ellipses and hyperbolas, their equations will start to feel like second nature.