IB myp 4-5 MATHEMATICS – Practice Questions- All Topics
Topic :Geometry-Coordinate Geometry
Topic :Geometry- Weightage : 21 %
All Questions for Topic : Metric conversions,Volume of regular polyhedra,Similarity and congruence,Coordinate geometry, including distance, midpoint and gradient formulae,Movement on a plane- isometric transformations, enlargements and tessellations,y=m x+c, gradients and intercepts (see also functions),Gradient of parallel lines,Circle geometry,Rotation around a given point
Question (6 Marks)
In the graph below, the function $f(x)=2 x+4$ intersects $g(x)=(x-1)^2-2$ at points $A$ and $B$.
Question (b) : 4 marks
Hence, find the coordinates $(x, y)$ of points $\mathrm{A}$ and $\mathrm{B}$.
▶️Answer/Explanation
Ans:
To find the coordinates $(x, y)$ of points $A$ and $B$, we can substitute the values of $x$ we found back into the equations of the functions $f(x) = 2x + 4$ and $g(x) = (x-1)^2 – 2$.
For point $A$ where $x = 5$:
$f(5) = 2(5) + 4 = 10 + 4 = 14$
$g(5) = (5-1)^2 – 2 = 4^2 – 2 = 16 – 2 = 14$
Therefore, the coordinates of point $A$ are $(5, 14)$.
For point $B$ where $x = -1$:
$f(-1) = 2(-1) + 4 = -2 + 4 = 2$
$g(-1) = (-1-1)^2 – 2 = (-2)^2 – 2 = 4 – 2 = 2$
Hence, the coordinates of point $B$ are $(-1, 2)$.
Therefore, the coordinates of points $A$ and $B$ are $(5, 14)$ and $(-1, 2)$, respectively.
Question (35 marks)
The straight line $T_1$ touches the parabola $y=k x^2$ once at point $A$ as shown in the simulation below where $k$ is the coefficient of $x^2$. As the value of $k$ changes, the point $A$ changes but the gradient of $T_1$ stays as 1 . Click on the arrow to see what happens to the coordinate $A$ as the value of $k$ changes.
Question (a) : 2 marks
As the value of $k$ changes, the following coordinates for point $\mathrm{A}$ are recorded in the table below. If you would like to add more values in the table, click inside the relevant box, then write the values in the “Add Label” box.
Describe in words a pattern for $x_A$.
▶️Answer/Explanation
Ans:
Based on the given table, we can observe a pattern for the $x$ coordinate of point $A$, denoted as $x_A$. The pattern indicates that the denominator of the fractions for $x_A$ is increasing by 2 as $k$ increases.
Specifically, for each value of $k$, the denominator of $x_A$ increases by 2 compared to the previous value. This pattern can be described as follows:
For $k = 1$, the denominator of $x_A$ is $2$.
For $k = 2$, the denominator of $x_A$ is $4$, which is $2$ more than the previous denominator of $2$.
For $k = 3$, the denominator of $x_A$ is $6$, which is $2$ more than the previous denominator of $4$.
This pattern continues as $k$ increases, with each denominator of $x_A$ being $2$ more than the previous denominator.
Therefore, the pattern for $x_A$ is that the denominator of the fraction increases by $2$ as $k$ increases.