Further Mathematics (9665) | Algebra and graphs

👋 Welcome to FP1: Algebra and Graphs!

Hello there! This chapter is your foundation for understanding how complex equations translate into beautiful, recognizable shapes on a graph. In Further Maths, we move beyond basic quadratics and straight lines to explore functions and curves that have fascinating properties, like asymptotes and foci.

Don't worry if some of the formulas look intimidating! We will break down the crucial skills: sketching these graphs accurately, finding their limits (asymptotes), and locating those tricky maximum and minimum points without relying on calculus.

Ready to transform algebra into geometry? Let's dive in!

1. Graphs of Rational Functions

A Rational Function is simply a fraction where the numerator and the denominator are both polynomials. We focus on three main forms in FP1:

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)

1.1 Finding Asymptotes

Asymptotes are invisible lines that the graph approaches but never touches (or sometimes crosses, but always approaches as \(x \to \infty\) or \(y \to \infty\)). They are crucial for sketching.

A. Vertical Asymptotes (VA)

These occur where the function is undefined, which means the denominator is equal to zero.

Step-by-step process:

1. Set the denominator equal to zero.
2. Solve for \(x\).
3. If the solution(s) for \(x\) are real, these are the equations of your vertical asymptotes.

Example: For \(y = \frac{x+5}{x-3}\), the VA is \(x-3=0\), so \(x=3\).

B. Horizontal Asymptotes (HA)

These describe the graph's behaviour when \(x\) gets very large (tends towards \(\pm\infty\)). To find the HA, look at the degree (highest power) of the polynomials in the numerator ($N$) and the denominator ($D$).

Think of it like a race:

• Case 1: Degree N = Degree D (They are running neck and neck)
  The HA is \(y = \frac{\text{Coefficient of highest power in N}}{\text{Coefficient of highest power in D}}\).
  Example: For \(y = \frac{2x^2+1}{5x^2-3x}\), the HA is \(y = \frac{2}{5}\). (This covers form 1 and 3 when the degrees match).
• Case 2: Degree N < Degree D (The denominator is much faster)
  The function value approaches zero. The HA is \(y = 0\) (the \(x\)-axis). (This covers form 2).

C. Non-Axial Asymptotes (Oblique/Slant Asymptotes)

Did you know? If the degree of the numerator is exactly one greater than the degree of the denominator (e.g., \(\frac{x^2}{x}\)), you get an asymptote that is a straight line, but not horizontal or vertical. Although this concept is generally important for rational functions, the syllabus specifies that asymptotes will always be parallel to the coordinate axes for the functions you are given in FP1.1. Therefore, you only need to focus on HA and VA!

1.2 Intersections and Sketching

To sketch the graph accurately, you need to find where the curve crosses the axes and any other specified straight lines.

1. Y-intercept: Set \(x=0\) and solve for \(y\).
2. X-intercept (Roots): Set \(y=0\). Since \(y = \frac{N(x)}{D(x)}\), this happens only when the numerator \(N(x)\) is zero (as long as \(D(x) \neq 0\)).
3. Intersection with line \(L\): If the line is \(y=mx+c\), set the function equal to the line equation and solve for \(x\).

1.3 Finding Maximum and Minimum Points (The FP1 Trick!)

For the function \(y = \frac{x^2+ax+b}{x^2+cx+d}\), finding the stationary points (maxima/minima) usually requires calculus. However, FP1 requires a clever method using quadratic theory.

This method finds the range of possible \(y\) values (i.e., the boundaries where the function exists).

Step-by-step for finding the range of \(y\) (or \(k\)):

Prerequisite: Remember that a quadratic equation \(Ax^2 + Bx + C = 0\) has real roots if and only if its discriminant \(\Delta = B^2 - 4AC \ge 0\).

1. Set \(y=k\) and cross-multiply: $$\frac{x^2+ax+b}{x^2+cx+d} = k \implies x^2+ax+b = k(x^2+cx+d)$$
2. Rearrange into a quadratic equation in \(x\): $$x^2(1-k) + x(a-ck) + (b-dk) = 0$$
3. Apply the Real Roots Condition: Since \(x\) must be a real number for the point \((x, k)\) to exist on the graph, the quadratic in \(x\) must have real roots. $$\Delta = (a-ck)^2 - 4(1-k)(b-dk) \ge 0$$
4. Solve the resulting inequality for \(k\): This inequality will be a quadratic in \(k\). Solving it gives the interval(s) where \(k\) (or \(y\)) can exist. The boundaries of this interval are the maximum and minimum values of the function.
5. Find the coordinates (optional but often required): To find the \(x\)-coordinates of these max/min points, substitute the boundary values of \(k\) back into the quadratic equation found in Step 2. Since the discriminant equals zero at these boundaries, the quadratic will have a repeated root \(x = -\frac{B}{2A}\).

Encouragement: This method is subtle! It connects the existence of a point on the graph (real \(x\)) to the algebraic condition for real solutions (\(\Delta \ge 0\)). Master this, and you've mastered the hardest part of this section.

1.4 Solving Associated Inequalities

You may need to solve inequalities like \(\frac{x^2+ax+b}{x^2+cx+d} > 0\), or \(\frac{ax+b}{cx+d} \le m\).

Common Mistake to Avoid: DO NOT multiply across by the denominator unless you are certain it is positive. This is dangerous because multiplying by a negative flips the inequality sign!

The Safe Method (Graphical/Critical Value Method):

1. Rearrange the inequality so that one side is zero. $$ \text{e.g., } \frac{ax+b}{cx+d} \le m \implies \frac{ax+b}{cx+d} - m \le 0 $$ Then combine the fractions: $$ \frac{(ax+b) - m(cx+d)}{cx+d} \le 0 $$
2. Find the Critical Values: These are the \(x\) values where the expression equals zero (numerator=0) or where it is undefined (denominator=0).
3. Sketch or Use a Sign Table: Plot the critical values on a number line. Test the sign of the expression in the regions between and outside these values.
4. Conclude: Select the regions that satisfy the original inequality.

Key Takeaway for Rational Functions: Focus on the algebraic structure (degrees of polynomials) to find asymptotes, and use the discriminant trick (\(\Delta \ge 0\)) to find maximum and minimum bounds.

2. Conic Sections: Parabolas, Ellipses, and Hyperbolas

Conic Sections are the curves formed when you slice a double cone with a plane. This section requires you to sketch these standard forms and understand how they interact with lines.

2.1 The Parabola

The standard equation of a parabola opening sideways is:

$$y^2 = 4ax$$

• Sketching: If \(a>0\), it opens to the right. If \(a<0\), it opens to the left. It is symmetrical about the \(x\)-axis.
• Vertex: (0, 0)
• Axis of Symmetry: \(y=0\) (the \(x\)-axis)

2.2 The Ellipse

An ellipse is like a stretched or squashed circle. Its standard equation is:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

• Sketching: The values \(a\) and \(b\) determine the semi-major and semi-minor axes.
• X-intercepts: \((\pm a, 0)\)
• Y-intercepts: \((0, \pm b)\)

Analogy: If \(a=b\), it becomes a circle! The ellipse is perfectly contained within a box defined by \(x=\pm a\) and \(y=\pm b\).

2.3 The Hyperbola

There are two primary standard forms for hyperbolas.

A. Standard Hyperbola (Opens horizontally)

$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$

• Sketching: The curve crosses the \(x\)-axis at \((\pm a, 0)\). It does not cross the \(y\)-axis (if we set \(x=0\), \(y^2\) must be negative).
• Asymptotes: These are crucial for sketching. They are defined by the straight lines: $$y = \pm \frac{b}{a}x$$

B. Rectangular Hyperbola

$$xy = c^2$$

If \(c^2 > 0\), the curve lies in the first and third quadrants (like \(y=1/x\), but scaled).

• Asymptotes: The axes themselves! \(x=0\) and \(y=0\).

2.4 Interpreting Intersections: The Discriminant Revisited

When finding the intersection points between a conic section (like a parabola or hyperbola) and a straight line \(y=mx+c\), you substitute the line equation into the conic equation. This results in a quadratic equation in \(x\).

The discriminant \(\Delta = B^2 - 4AC\) of this resulting quadratic has a clear geometrical meaning:

• \(\Delta > 0\) (Distinct Real Roots): The line intersects the conic at two distinct points.
• \(\Delta = 0\) (Equal Real Roots): The line is a tangent to the conic, intersecting at exactly one point.
• \(\Delta < 0\) (No Real Roots): The line does not intersect the conic at all.

2.5 Transformations of Conic Sections

You must know the effect of applying single transformations to these equations. Remember, the transformation affects the coordinates \((x, y)\) and you replace them in the original equation.

Transformation	Effect on Coordinates	Substitution in Equation
Translation by vector \(\begin{pmatrix} p \\ q \end{pmatrix}\)	\((x, y) \to (x+p, y+q)\)	Replace \(x\) with \((x-p)\) Replace \(y\) with \((y-q)\)
Stretch, scale factor \(k\), parallel to x-axis	\((x, y) \to (kx, y)\)	Replace \(x\) with \((x/k)\)
Stretch, scale factor \(k\), parallel to y-axis	\((x, y) \to (x, ky)\)	Replace \(y\) with \((y/k)\)
Reflection in the line \(y=x\)	\((x, y) \to (y, x)\)	Swap \(x\) and \(y\) in the equation.

Example: If the ellipse \(\frac{x^2}{4} + \frac{y^2}{9} = 1\) is translated by \(\begin{pmatrix} 1 \\ -2 \end{pmatrix}\), the new equation is \(\frac{(x-1)^2}{4} + \frac{(y+2)^2}{9} = 1\).

Key Takeaway for Conics: Always identify \(a\) and \(b\). Use these constants to find intercepts and asymptotes (for hyperbolas). The discriminant links the algebra of intersections directly to the geometry of the curve.

Summary and Key Takeaways

The FP1 'Algebra and Graphs' chapter is fundamentally about linking algebraic manipulation to graphical interpretation. Your success depends on two non-calculus techniques:

1. Asymptotes: Using ratios of coefficients (Horizontal) or zero denominators (Vertical).
2. Range/Max/Min: Using the quadratic discriminant \(\Delta \ge 0\) by rearranging \(y=k\) into a quadratic in \(x\).
3. Intersections: Interpreting \(\Delta\) for conics to distinguish between two intersections, a tangent, or no intersections.

Keep practicing these core skills, and you'll find that these sophisticated curves are highly predictable!