Common function graphs
There are some common function graph shapes that you should know about.
This page lists some of them.
Common graph terminology
- The y-intercept is the point where the graph crosses the y-axis (where
x=0 ). - The x-intercepts (or roots) are the points where the graph crosses the
x-axis (where
y=0 ). - Turning points are points where the graph changes from increasing (positive gradient) to decreasing (negative gradient), or the other way round. It is also sometimes called the vertex. The gradient of the line at a turning point is 0.
- An asymptote is a line that the graph approaches as
x gets infinitely big or small, but never touches.
Quadratics
- Quadratics are the shape of a parabola.
- They have the general form of
y=ax^2 + bx + c . - The line that goes through the vertex and divides the graph into two symmetrical halves is called the axis of symmetry.
- This is because parabolas are symmetrical (with the line of symmetry going through the vertex).
- The y-intercept is at the point
(0, c) .
Cubics
- Cubics have the general form of
y=ax^3 + bx^2 + cx + d . - They can have up to 2 turning points.
- The y-intercept is at the point
(0, d) . - Cubics can have 1, 2, or 3 x-intercepts (roots).
Quartics
- Quartics have the general form of
y=ax^4 + bx^3 + cx^2 + dx + e . - They can have up to 3 turning points.
- The y-intercept is at the point
(0, e) . - They can have 1, 2, 3, or 4 x-intercepts (roots).
Exponentials
- Exponential functions have the general form of
y=ab^x , whereb>0 andb \neq 1 . - They have a horizontal asymptote at
y=0 (the x-axis). - The y-intercept is at the point
(0, a) .
Reciprocals
- Reciprocal functions have the general form of
y=\frac{a}{x} . - They have two asymptotes: a vertical asymptote at
x=0 (the y-axis) and a horizontal asymptote aty=0 (the x-axis). - The graph is in two separate parts (called branches), one in the first and third
quadrants if
a>0 , and one in the second and fourth quadrants ifa<0 . - There is no y-intercept or x-intercept, since the graph never touches either axis.
flashcards
| Question | Answer |
|---|---|
| What is the y-intercept of a graph? | The point where the graph crosses the y-axis (where |
| What are x-intercepts (or roots)? | The points where the graph crosses the x-axis (where |
| What is a turning point (or vertex)? | A point where the graph changes from increasing (positive gradient) to decreasing (negative gradient), or vice versa; the gradient at this point is 0. |
| What is an asymptote? | A line that the graph approaches as |
| What shape do quadratic functions have? | The shape of a parabola. |
| What is the general form of a quadratic? | |
| What is the axis of symmetry of a quadratic? | The line that goes through the vertex and divides the graph into two symmetrical halves. |
| Why are parabolas described as symmetrical? | They have a line of symmetry going through the vertex. |
| Where is the y-intercept of a quadratic? | At the point |
| What is the general form of a cubic? | |
| What is the maximum number of turning points a cubic can have? | Up to 2 turning points. |
| Where is the y-intercept of a cubic? | At the point |
| How many x-intercepts (roots) can a cubic have? | 1, 2, or 3. |
| What is the general form of a quartic? | |
| What is the maximum number of turning points a quartic can have? | Up to 3 turning points. |
| Where is the y-intercept of a quartic? | At the point |
| How many x-intercepts (roots) can a quartic have? | 1, 2, 3, or 4. |
| What is the general form of an exponential function? | |
| What is the horizontal asymptote for the exponential | |
| Where is the y-intercept of the exponential | At the point |
| What is the general form of a reciprocal function? | |
| What asymptotes does the reciprocal | A vertical asymptote at |
| Where are the two branches of | In the first and third quadrants. |
| Where are the two branches of | In the second and fourth quadrants. |
| Does | No, it never touches either axis, so there is no y-intercept or x-intercept. |