• Home

Parabolas

Aquadratic functionis a function that can be written in the form$f\left(x\right)=a{x}^2+bx+c$where$a,b$, and$c$are real numbers and$a\ne 0$. This form is called the standard form of a quadratic function.

Thegraph二次函数的u型曲线是called aparabola.

The graph of the equation$y=x^2$, shown below, is a parabola. (Note that this is a quadratic function in standard form with$a=1$and$b=c=0$.)

In the graph, the highest or lowest point of a parabola is the vertex. The vertex of the graph of$y=x^2$is$\left(0,0\right)$.

If$a>0$in$f\left(x\right)=a{x}^2+bx+c$, the parabolaopens upward. In this case the vertex is the minimum, or lowest point, of the parabola. A large positive value of$a$makes a narrow parabola; a positive value of$a$which is close to$0$makes the parabola wide.

If $a<0$in$f\left(x\right)=a{x}^2+bx+c$, the parabolaopens downward. In this case the vertex is the maximum, or highest point, of the parabola. Again, a large negative value of$a$makes the parabola narrow; a value close to zero makes it wide.

For an equation in standard form, the value of$c$gives the $y$-interceptof the graph.

The line that passes through the vertex and divides the parabola into two symmetric parts is called the axis of symmetry.

The equation of the axis of symmetry for the graph of$y=a{x}^2+bx+c$, where$a\ne 0$, is$x=-\frac{b}{2a}$

In all the above graphs, the axis of symmetry is the$y$-axis,$x=0$. In the graphs below, the axis of symmetry is different (marked in red.) Note that$c$still gives the$y$-intercept.

If you write a quadratic function like$x=f\left(y\right)=a{y}^2+by+c$, where$x$is a function of$y$(instead of a$y$as a function of$x$), you get a parabola where the axis of symmetry is horizontal.

Note that in this case,$c$is the$x$-intercept. If$a$is positive, the graph opens to the right; if$a$is negative, the graph opens to the left.

See also

Axis of Symmetry of a Parabola

Vertex of a Parabola

Graphing quadratic equations using the axis of symmetry