Test of Mathematics for University Admission (TMUA) Coordinate Geometry and Graphs Syllabus

Coordinate Geometry and Graphs syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Coordinate Geometry and Graphs in Test of Mathematics for University Admission (TMUA), not a summary of it.

1. 1

   Straight Lines

   3 topics

   • Gradient and Equation of a Line
     • Forms y=mx+c and ax+by+c=0
     • Line through two points
   • Parallel and Perpendicular Lines
     • Gradient conditions
     • Perpendicular bisectors
   • Distance and Midpoint
     • Length of a line segment
     • Midpoint coordinates
2. 2

   Circles

   3 topics

   • Equation of a Circle
     • Centre-radius form
     • General expanded form and completing the square
   • Circle Geometry Properties
     • Tangent perpendicular to radius
     • Angle in a semicircle
   • Intersections with Lines and Circles
     • Tangency conditions
     • Chord and secant problems
3. 3

   Graphs of Functions

   3 topics

   • Standard Curve Sketching
     • Linear, quadratic and cubic curves
     • Reciprocal and rational curves
     • Exponential and logarithmic curves
   • Intercepts and Asymptotes
     • Axis crossings and behaviour at infinity
     • Vertical and horizontal asymptotes
   • Graphical Solution of Equations
     • Intersection interpretation
     • Counting roots from graphs
4. 4

   Transformations of Graphs

   4 topics

   • Translations
     • Horizontal and vertical shifts
   • Stretches
     • Scaling parallel to axes
   • Reflections
     • Reflection in the axes
   • Combined Transformations
     • Order of composite transformations

Coordinate Geometry and Graphs flashcards for Test of Mathematics for University Admission (TMUA)

22 of 50 cards from the Coordinate Geometry and Graphs deck — real questions with worked answers.

1. What is the gradient of the line joining points $(x_{1},y_{1})$ and $(x_{2},y_{2})$?

   $$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$ It is the change in $y$ divided by the change in $x$ (rise over run).

2. State the gradient–intercept form of the equation of a straight line and name each parameter.

   $$y=mx+c$$ where $m$ is the gradient and $c$ is the $y$-intercept (the value of $y$ when $x=0$).

3. Give the point–slope form of a line through $(x_{1},y_{1})$ with gradient $m$.

   $$y-y_{1}=m(x-x_{1})$$

4. What does the general form $ax+by+c=0$ give for the gradient of the line?

   $$m=-\frac{a}{b}$$ (valid when $b\neq 0$).

5. How are the gradients of two parallel lines related?

   They are equal: if the lines are parallel then $m_{1}=m_{2}$.

6. What is the condition on gradients for two lines to be perpendicular?

   Their gradients multiply to $-1$: $$m_{1}m_{2}=-1,\qquad m_{2}=-\frac{1}{m_{1}}$$ (the perpendicular gradient is the negative reciprocal).

7. A line has gradient $\frac{3}{4}$. What is the gradient of any line perpendicular to it?

   $$-\frac{4}{3}$$ (the negative reciprocal).

8. What can you say about the gradient of a horizontal line and a vertical line?

   A horizontal line has gradient $0$ (equation $y=k$). A vertical line has an undefined (infinite) gradient (equation $x=k$).

9. State the distance formula between points $(x_{1},y_{1})$ and $(x_{2},y_{2})$.

   $$d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$$

10. State the midpoint formula for the segment joining $(x_{1},y_{1})$ and $(x_{2},y_{2})$.

    $$M=\left(\frac{x_{1}+x_{2}}{2},\ \frac{y_{1}+y_{2}}{2}\right)$$

11. What is the equation of the perpendicular bisector of a segment, and how do you find it?

    It passes through the segment's midpoint and is perpendicular to the segment. Find the midpoint and the segment's gradient $m$, then use gradient $-\frac{1}{m}$ through the midpoint in $y-y_{1}=-\frac{1}{m}(x-x_{1})$.

12. Give the standard equation of a circle with centre $(a,b)$ and radius $r$.

    $$(x-a)^{2}+(y-b)^{2}=r^{2}$$

13. Write the equation of a circle centred at the origin with radius $r$.

    $$x^{2}+y^{2}=r^{2}$$

14. Convert the general circle equation $x^{2}+y^{2}+2gx+2fy+c=0$ into centre and radius form.

    Centre $(-g,-f)$ and radius $$r=\sqrt{g^{2}+f^{2}-c}$$ (provided $g^{2}+f^{2}-c>0$).

15. How do you find the centre and radius of a circle given in expanded form, e.g. $x^{2}+y^{2}-6x+4y-3=0$?

    Complete the square in $x$ and $y$: $(x-3)^{2}+(y+2)^{2}=16$, giving centre $(3,-2)$ and radius $4$.

16. Circle property: what is the relationship between the angle in a semicircle and the diameter?

    The angle in a semicircle is a right angle ($90^{\circ}$). Equivalently, the angle subtended by a diameter at any point on the circle is $90^{\circ}$.

17. Circle property: how does a tangent meet the radius at the point of contact?

    The tangent to a circle is perpendicular to the radius drawn to the point of contact (they meet at $90^{\circ}$).

18. Circle property: what is the relationship between the perpendicular from the centre and a chord?

    The perpendicular from the centre to a chord bisects the chord (and conversely, the perpendicular bisector of a chord passes through the centre).

19. How do you find the equation of the tangent to a circle at a given point on it?

    Find the gradient of the radius from the centre to that point, take its negative reciprocal for the tangent gradient, then use the point–slope form through the point of contact.

20. How can you determine whether a point $(x_{0},y_{0})$ lies inside, on, or outside a circle of centre $(a,b)$ radius $r$?

    Compare $(x_{0}-a)^{2}+(y_{0}-b)^{2}$ with $r^{2}$: less than $r^{2}$ means inside, equal means on the circle, greater means outside.

21. To find where a line meets a circle, what method do you use and what does the discriminant tell you?

    Substitute the line equation into the circle equation to get a quadratic in one variable. The discriminant $b^{2}-4ac$: $>0$ two intersection points (secant), $=0$ one point (tangent), $<0$ none (line misses circle).

22. How do you find the intersection points of two curves/lines algebraically?

    Solve their equations simultaneously (by substitution or elimination); each solution pair $(x,y)$ is an intersection point. For a line and curve, substitute to form one equation in a single variable.

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