Curved Graphs Quiz
Total: 40 marks
Answer all questions. Show working for Questions 5–8. Give exact answers where appropriate.
1. [1 mark]
A quadratic function has its vertex at $(2,-3)$ and passes through $(4,1)$. Which equation represents the function?
A. $y=(x-2)^2-3$
B. $y=(x-2)^2+3$
C. $y=2(x-2)^2-3$
D. $y=(x+2)^2-3$
2. [1 mark]
What type of graph is represented by $y=\frac{1}{x}$?
A. A straight line
B. A reciprocal graph with two branches
C. An exponential decay graph
D. A parabola
3. [1 mark]
What is the axis of symmetry of the parabola $y=x^2-6x+9$?
A. $x=-3$
B. $x=3$
C. $x=0$
D. $x=6$
4. [1 mark]
Which statement about a tangent to a curve at a point is correct?
A. It always crosses the curve at exactly two points.
B. It is always parallel to the $x$-axis.
C. It has the same gradient as the curve at the point of contact.
D. It is always perpendicular to the $y$-axis.
5. [4 marks]
For the quadratic function $y=x^2-6x+5$:
- Find the equation of its axis of symmetry.
- Find the coordinates of its turning point.
- State whether the turning point is a maximum or a minimum.
6. [4 marks]
The graph has equation $y=-\frac{3}{x}$.
- State its vertical and horizontal asymptotes.
- State which two quadrants contain its branches.
- Explain why the graph does not cross either asymptote.
7. [7 marks]
Consider the quadratic function
$$f(x)=x^2-4x+1.$$
- Complete the square to write $f(x)$ in the form $(x-h)^2+k$.
- Find the coordinates of the turning point and state whether it is a maximum or a minimum.
- Find the $x$-intercepts in exact form.
- Find the $y$-intercept.
8. [11 marks]
Consider the cubic function
$$f(x)=x^3-3x^2-9x+5.$$
- Find $f’(x)$ and solve $f’(x)=0$ to find the $x$-coordinates of the turning points.
- Find the coordinates of both turning points.
- Use the second derivative to classify each turning point as a local maximum or local minimum.
- State the end behaviour of the graph as $x\to-\infty$ and as $x\to+\infty$.
9. [4 marks]
For the curve $y=3x^4+2x^3-5x+7$:
- Find $\frac{dy}{dx}$.
- Find the gradient of the curve at $x=1$.
Attempt the quiz before checking the mark scheme
Mark Scheme M18 - Curved Graphs Mark Scheme
Topic guide: M18 - Curved Graphs