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Mathematics for IGCSE & O level - Sketching Curves (Section 4)

1
What is the shape of the graph of a function of the form \(y = \frac{1}{x}\)?
Answer
(C)
A hyperbola
2
In the general form of a quadratic equation, \(y = ax^2 + bx + c\), what does the constant 'c' represent?
Answer
(B)
The y-intercept of the graph.
3
Which of the following is a characteristic of a graph of the form \(y=x^3\)?
Answer
B
D
4
If the function is \(y = x^2 + 4x + 4\), at what x value does the graph intersect the x-axis?
Answer
(D)
-2
5
How are the roots of a quadratic equation related to the graph of the equation?
Answer
(B)
The roots are the x-intercepts of the graph.
6
Which form of a quadratic function allows you to easily identify the coordinates of the turning point?
Answer
(C)
Completed Square Form
7
What type of graph has a 'double bend' in its basic shape?
Answer
(C)
Cubic
8
If the turning point of a quadratic graph is at (2, 3) and the line of symmetry is x = 2, what is the equation of the graph's axis of symmetry?
Answer
(C)
x = 2
9
For which type of graph would it be useful to complete the square?
Answer
(B)
Quadratic
10
Which form of the quadratic equation can quickly identify the x-intercepts (roots)?
Answer
(C)
Factored Form
11
What does the term 'turning point' refer to on a quadratic graph?
Answer
(C)
The vertex, where the graph changes direction.
12
Which of the following functions is characterized by two asymptotes?
Answer
(C)
Reciprocal
13
What is a defining characteristic of a cubic graph?
Answer
B
C
D
14
Which of the following is true about the asymptotes of a reciprocal graph of the form \(y = \frac{a}{x-h} + k\)?
Answer
(A)
Vertical asymptote at x=h and horizontal asymptote at y=k.
15
Which of the following are true about sketching curves?
Answer
A
B
16
When sketching the cubic graph \(y = x^3\), where does the graph pass through?
Answer
(A)
(0, 0) and (1, 1)
17
Which of the following is a characteristic of a hyperbola?
Answer
(B)
Two asymptotes
18
In the function \(y = x^3 - 8\), where would the graph cross the x-axis?
Answer
(B)
x = 2
19
What is the purpose of the line of symmetry in a parabola?
Answer
(B)
It cuts the graph in half, reflecting each side.
20
If the graph of \(y=ax^2+bx+c\) touches the x-axis at only one point, what is the x-coordinate called?
Answer
(B)
The x-intercept, or root.
21
How do you find the x-intercept(s) of a function?
Answer
B
C
22
What part of the quadratic function is used to find the axis of symmetry?
Answer
B
C
23
Which of the following is a property of exponential graphs of the form \(y = a^x\) with a > 1?
Answer
A
B
C
24
What is the y-intercept of the graph of the equation \(y = 2^x - 1\)?
Answer
(A)
0
25
What is the sign of 'a' in a quadratic equation \(ax^2 + bx + c = y\) if the graph opens downwards?
Answer
(B)
a < 0
26
The graph of a cubic function is symmetrical around what point?
Answer
(C)
The origin
27
What is the horizontal asymptote of the graph of \(y = 2^{x-1} + 3\)?
Answer
(B)
y = 3
28
When sketching curves, which piece of information is the most important to get correct?
Answer
(C)
The shape and asymptotes.
29
What will happen to the graph of \(y=x^2\) if we change it to \(y=(x-3)^2+2\)?
Answer
(B)
Shift 3 units right and 2 units up
30
What is the turning point of a quadratic graph that is expressed in the form \(y = a(x - p)^2 + q\)?
Answer
(B)
(p, q)
31
Which is the equation for the line of symmetry of \(y=x^2 - 4x + 7\)?
Answer
(A)
x = 2
32
Which of the following is true regarding the graph of \(y=x^3-x\)?
Answer
(C)
The graph passes through the origin.
33
What type of graph is symmetrical around the y-axis?
Answer
(B)
Quadratic (when the x-value of the turning point is zero)
34
What is the horizontal asymptote for the graph \(y=3^x\)?
Answer
(D)
y = 0
35
In a quadratic equation, what is the effect of a negative 'a' value?
Answer
(B)
The parabola opens downwards.
36
In a quadratic graph, what is the relationship between the x-coordinate of the vertex and the line of symmetry?
Answer
(B)
The line of symmetry is the vertical line at x = x-coordinate
37
For the graph of \(y = \frac{1}{x + 5} - 1\), what are the equations of the asymptotes?
Answer
(A)
x = -5, y = -1
38
If \(y= 3x^2 + 6x + 2\) is transformed to the vertex form, what is the x-coordinate of the vertex?
Answer
(C)
-1
39
Which is a characteristic of \(y = 2^{-x}\) compared to \(y = 2^x\)?
Answer
(A)
Reflected over the y-axis
40
How would you describe the behavior of a graph where y = a^x when a < 0?
Answer
(A)
The graph does not exist.
41
What does the 'a' represent in the quadratic equation \(y = ax^2 + bx + c\)?
Answer
(C)
The direction of opening and stretch.
42
What is true about the graph of \(y = \frac{1}{x} + 2\)?
Answer
(B)
It has horizontal asymptote at y=2
43
When sketching a quadratic graph of the form \(y = ax^2 + bx + c\), which steps are usually needed?
Answer
A
B
C
44
What is a key feature when sketching a graph?
Answer
A
B
D
45
For a reciprocal graph of the form \(y = \frac{1}{x}\), what are the equations of the asymptotes?
Answer
(A)
x = 0 and y = 0
46
What happens to the graph of \(y = \frac{1}{x}\) if the equation becomes \(y = \frac{1}{x} + 4\)?
Answer
(C)
The graph shifts 4 units upwards.
47
What is the y-intercept of \(y=(x+3)(x-1)\)?
Answer
(C)
-3
48
Which type of graph has a horizontal asymptote?
Answer
(D)
Exponential
49
If the function \(y=x^3-27\) intersects the x-axis, what is the x-coordinate of the point of intersection?
Answer
(A)
3
50
If the function is \(y = -x^2 + 4\), find the x-intercepts.
Answer
(B)
-2, 2