Unit 2 - Functions
Required Prior Knowledge
Questions
Solve the equation$$\frac{x^{2}+5x}{x+1}=3$$
Solutions
Get Ready
Questions
Sketch the graphs of \(y=x-3\) and \(2x+3y=16\).
Find the point of intersection of these two linear functions.
How many different ways could you do this?
Solutions
Notes
The intersection points of two functions \(f\left(x\right)\) and \(g\left(x\right)\) are the points \(\left(x,y\right)\) where the graphs meet. At these points, the \(x\) values and corresponding \(y\) values are the same for both functions.
To find the points of intersection, we set the functions equal to each other and solve the resulting equation:$$f\left(x\right)=g\left(x\right)$$This gives the \(x\) coordinate(s) of the point(s) of intersection, and then we substitute these into either\(f\left(x\right)\) or \(g\left(x\right)\) to find the \(y\) coordinate(s).
Consider the graph of a quadratic function and a linear function.
There are 3 possible scenarios:
The line intersects the quadratic two times.
The line just touches the quadratic in one place.
The line does not intersect the quadratic.
When a line touches a curve we call it a tangent.
We can solve the following pairs of functions ‘by hand’:
Linear and linear
Linear and Quadratic
Quadratic and Quadratic
Cubic and linear/quadratic in simple cases
For any other combinations (including exponentials for example), you will probably use your GDC to find the points of intersection.
It is common in this course to be expected to find these points of intersection with the GDC.
Examples and Your Turns
Example
Find the point(s) of intersection of the graphs with equations \(y=x^{2}-x-18\) and \(y=x-3\), if they exist.
How could we determine how many points of intersection there are, without solving the equation?
Your Turn
Find the point(s) of intersection of the graphs with equations \(y=x^{2}+5x+6\) and \(y=2x^{2}+2x-4\), if they exist.
Hence determine when \(x^{2}+5x+6\gt 2x^{2}+2x-4\).
Example
\(y=2x+k\) is a tangent to \(y=2x^{2}-3x+4\). Find \(k\).
Your Turn
\(y=-4x+k\) is a tangent to \(y=3x^{2}+3x-5\). Find \(k\).
Your Turn
Find the values of \(m\) for which the line \(y=mx-5\) is a tangent to the curve \(y=x^{2}+3x+4\).
Your Turn
Given that the straight line \(y=3x+c\) is a tangent to the curve \(y=x^{2}+9x+k\), express \(k\) in terms of \(c\).
Your Turn
Find the exact value of \(m\) for which the line \(y=mx+5\) is a tangent to the curve \(x^{2}+y^{2}=10\).
Your Turn
Find the values of \(k\) for which the line \(y+kx-2=0\) is a tangent to the curve \(y=2x^{2}-9x+4\).
Your Turn
Find the values of \(k\) for which the line \(y=2x+k+2\) cuts the curve \(y=2x^{2}+\left(k+2\right)x+8\) in two distinct points.
Your Turn
Find the set of values of \(k\) for which the line \(y=k\left(4x-3\right)\) does not intersect the curve \(y=4x^{2}+8x-8\).
Your Turn
Find the values of \(k\) for which the line \(y=kx-3\) does not meet the curve \(y=2x^{2}-3x+k\).
Your Turn
Consider the functions \(f\left(x\right)=e^{x}\) and \(g\left(x\right)=-x+3\). Use your GDC to find the coordinates of any points of intersection. Give your answer(s) to 3 significant figures.
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
If two functions intersect, what is true about the functions at the point of intersection?
Describe the different ways that two quadratic functions could intersect. Give examples of each type.
Common Mistakes / Misconceptions
Forgetting to find the \(y\) coordinate after solving for \(x\).
Algebraic errors in manipulating the expressions to solve the equation.
Connecting This to Other Skills
This skill builds upon straight line graphs (2.1) and quadratic functions (2.7), and requires the skill of solving quadratic equations (PK3).
You also need to understand solving quadratic inequalities (2.15) and the discriminant (2.16).
We will link this idea to solving more complicated equations (2.23). It will also be essential when we get to solving integrals between two curves (5.13).
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?