Key Features of Graphs
Maximum and Minimum Values
Maximum and minimum values are points that represent the highest and lowest y-values of a function within a domain. These points are useful for solving optimisation problems.
ExampleConsider the quadratic function $f(x) = -x^2 + 4x + 5$. Using a graphing calculator, we can determine that the maximum point occurs at (2, 9). This means the highest y-value the function reaches is 9, occurring when x = 2.
TipWhen using technology to find max/min values, always check if they are global (absolute) or local (relative) extrema. Some functions may have multiple local extrema but only one global maximum or minimum.
Intercepts
Intercepts are the points where a graph crosses the $x$ or $y$-axis.
- X-intercepts (or roots) occur where $y = 0$
- Y-intercepts occur where $x = 0$
For the function $g(x) = x^2 - 4x + 3$, using a graphing calculator reveals:
- $x$-intercepts: (1, 0) and (3, 0)
- $y$-intercept: (0, 3)
Alternatively, it should be obvious that
$$g(0)=3$$
and
$$g(x) = (x-3)(x-1)$$
So the $x$ and $y$ intercepts can also be found analytically.
NoteIntercepts are particularly useful in analyzing the behavior of functions and solving equations graphically.
Symmetry
Symmetry in graphs can be of two types:
- Symmetry about the y-axis: $f(x) = f(-x)$.
- Symmetry of rotation by 180 degrees about the origin: $f(x) = -f(-x)$.
Identifying symmetry can simplify the analysis of functions and help in sketching graphs more accurately.
ExampleThe function $h(x) = x^4 - 2x^2$ is symmetric about the y-axis. This can be verified by graphing and observing that the left and right halves of the graph are mirror images of each other.
Vertical and Horizontal Asymptotes
Asymptotes are lines that a graph approaches but never reaches.
- Vertical asymptotes occur where the function is undefined (often where the denominator equals zero in rational functions).
- Horizontal asymptotes show the behavior of the function as x approaches infinity or negative infinity.
The function $m(x) = \frac{2x+1}{x-3}$ has:
- A vertical asymptote at x = 3
- A horizontal asymptote at y = 2
These can be visualized clearly using a graphing calculator.
Using Graphing Technology
Graphing calculators allow students to quickly plot functions, zoom in on specific regions, and calculate key features with precision.
TipMost graphing calculators have built-in functions to find zeros, intersections, and extrema. Familiarize yourself with these features to save time during exams.
Finding Intersections
To find the intersection of two curves or lines:
- Graph both functions on the same coordinate plane.
- Use the calculator's intersection feature to find the exact coordinates where the graphs cross.
To find the intersection of $y = x^2$ and $y = 2x - 1$:
- Graph both functions
- Use the intersection feature
- The calculator shows intersections at approximately (-0.414, 1.828) and (2.414, 3.828)
Always check for multiple intersection points, as some function pairs may intersect more than once.
Exploring Effects of Parameters
Using technology with sliders allows for dynamic exploration of how changing parameters affects graph shapes and key features.
ExampleConsider the family of functions $f(x) = ax^2 + bx + c$:
- Varying 'a' changes the opening direction and width of the parabola
- Changing 'b' shifts the vertex horizontally
- Adjusting 'c' moves the entire graph up or down
Using a graphing tool with sliders for a, b, and c allows for real-time visualization of these effects.
TipWhen exploring parameter effects, focus on one parameter at a time to clearly understand its specific impact on the graph.
