Chapter Test
Sign In
The simplest quadratic function is y=x^2, while a general quadratic function is of the form f(x) = ax^2+bx+c. Try different values of the parameters a, b, and c. How does the graph changes with respect to the simpler function's graph?
See solution.
Let's start by reviewing the simplest quadratic function, y=x^2. Its graph is a parabola, with its vertex at the origin and its axis of symmetry (AOS) is the y-axis.
However, in general, a quadratic function is of the form shown below. f(x) = ax^2+bx+c This is known as the standard form of a quadratic function. Let's explore the effects of the parameters a, b, and c and what information we can get from them.
Since the parameter a multiplies the x^2-term, it can stretch or compress the parabola vertically.
You can explore this below. Give it a try!
If the value is negative, the same behavior holds but the graph would be upside down. The parabola is reflected across the x-axis.
Note that when x= 0, the function takes the form f( 0)= c. Therefore, c is the y-intercept of the quadratic function.
The AOS is the line x= - b2a. Moreover, since the AOS passes through the vertex, the x-coordinate of the vertex is = - b2a. The y-coordinate can be found by evaluating the quadratic function at that x-value Axis of Symmetry 1.5cm Vertex 0.75cm 0.65cmx= - b/2a 1.35cmV = ( - b/2a , f( - b/2a ) ) For example, consider the quadratic equation f(x) = x^2+2x+3. We can compare it to the general form of a quadratic equation to identify the parameters needed. f(x) = ax^2+ bx+ c f(x) = 1x^2+ 2x+ 3 As we can see, for this case a= 1 and b= 2. Let's calculate the equation for the AOS.
As we mentioned before this is also the x-coordinate of the vertex. Now let's find the y-coordinate.
x= - 1
Calculate power
Multiply
Add and subtract terms
Therefore the vertex is located at (- 1, 2).
Let's summarize the information we discussed above. The graph of any quadratic function is a parabola and its standard form is f(x) =ax^2+bx+c.