Interactive Coursebook (Online) of C12 Curve Sketching
C12 Graphs of Functions (II)
What’s on the Specification
| MM1.7 | Qualitative understanding that a function is a many‑to‑one (or sometimes just a one‑to‑one) mapping. Familiarity with the properties of common functions, including $f(x)=\sqrt{x}$ (which always means the ‘positive square root’) and $f(x)=|x|$. |
| MM6.3 | Applications of differentiation to gradients, tangents, normals, stationary points (maxima and minima only), strictly increasing functions [if $f^{\prime}(x)>0$] and strictly decreasing functions [if $f^{\prime}(x)<$ $0$]. Points of inflexion will not be examined, although a qualitative understanding of points of inflexion in the curves of simple polynomial functions is expected. |
| MM8.1 | Recognise and be able to sketch the graphs of common functions that appear in this specification: these include lines, quadratics, cubics, trigonometric functions, logarithmic functions, exponential functions, square roots, and the modulus function. |
| MM8.2 | Knowledge of the effect of simple transformations on the graph of y = f(x) with positive or negative value of a as represented by:
Compositions of these transformations. Knowledge and use of the notation $f(g(x))$. |
| MM8.3 | Understand how altering the values of $m$ and $c$ affects the graph of $y=m x+c$. |
| MM8.4 | Understand how altering the values of $a$, $b$ and $c$ in $y=a(x+$ $b)^2+c$ affects the corresponding graph. |
| MM8.5 | Use differentiation to help determine the shape of the graph of a given function, including:
|
| MM8.6 | Use algebraic techniques to determine where the graph of a function intersects the coordinate axes; appreciate the possible numbers of real roots that a general polynomial can possess. |
| MM8.7 | Geometric interpretation of algebraic solutions of equations; relationship between the intersections of two graphs and the solutions of the corresponding simultaneous equations. |
Learning Objectives
- Evaluate formal mappings and the modulus by distinguishing strictly between one-to-one and many-to-one relations, and accurately sketching functions defined by principal square roots $(\sqrt{x})$ and absolute values $(y=|f(x)|$ and $y=f(|x|))$
- Synthesise calculus with curve anatomy by utilising the first derivative $\left(f^{\prime}(x)\right)$ to dynamically map a curve, locating tangents, normals, and stationary points, and defining strict intervals where a function is increasing $\left(f^{\prime}(x)>0\right)$ or decreasing $\left(f^{\prime}(x)<0\right)$.
- Execute composed transformations by combining advanced algebraic manipulations, including vertical and horizontal stretches ( $y=a f(x)$ and $y=f(a x)$ ), to accurately track the shifting of algebraic roots and turning points.
- Sketch complex rational and logarithmic functions by extending core curve architectures to include advanced trigonometric bounds, utilising algebraic division and limit logic to determine exact asymptotes.
Suggested Time
| Task | No. of Questions | Suggested Time |
|---|---|---|
| Examples | 17 | 180 min |
| Quiz | 9 | 20 min |
| Practice A | 30 | 75 min |
| Practice B | 20 | 75 min |
| Total | 76 | approx. 5-6 hrs |
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