Quadratic functions involve a variable being squared. The graphs of quadratic functions are parabolas that are usually transformed versions of the basic equation \(y = x^2\). Graphing quadratic functions give us more information about the x-intercepts, which are also the solutions to quadratic equations. We can also know the number of solutions to a quadratic equation by graphing it.
Quadratic Function: A nonlinear function that can be written as \(y = ax^2 + bx + c\), where \(a ≠ 0\).
Parabola: The U-shaped graph formed by a quadratic function.
Vertex: The lowest or highest point on the parabola.
Axis of Symmetry: The vertical line that divides the parabola into two symmetric parts (mirror images).
x-Intercept: The point where a line crosses the x-axis.
Quadratic Equation: A nonlinear equation that can be written as \(ax^2 + bx + c = 0\) where \(a ≠ 0\).
Root (of a polynomial): The value that makes the polynomial equal to \(0\).
Zero (of a function): The value that makes the function equal to \(0\).
The standard form of a quadratic function: \(y = ax^2 + bx + c, a ≠ 0\).
When graphed, all quadratic functions are parabolas. The values of \(a\), \(b\), and \(c\) affect how the parabola looks.
The most straightforward way to graph quadratic functions is to make a table of values, plot them, then connect the points with a smooth curve.
Another way to graph quadratic functions is to rewrite the function in the intercept form.
The intercept form is just the factored form of the quadratic function.
The intercept form is useful because it tells us:
With these points, we can draw the parabola.
Knowing the x-intercepts and one other point on the parabola, we can also write the quadratic function for the parabola.
Finding the roots or zeroes of a quadratic equation is the same as solving the quadratic equation; we want to find the values of \(x\) that make the equation equal to \(0\).
A quadratic equation is the same as a quadratic function where \(y = 0\).
The x-intercepts can be found by factoring the equation (rewritten in the intercept form) or by graphing.
Looking at the graph tells us the number of solutions to the quadratic equation.