We will start by rewriting the function in intercept form. To do so, we will factor the right-hand side of the given equation.
Identify and Plot the x-intercepts
Recall the intercept form of a quadratic function.
f(x)=a(x-p)(x-q)
In this form, where a ≠ 0, the x-intercepts are p and q. Let's consider the intercept form of our function.
f(x)=2(x-2)^2 ⇔ f(x)= 2(x- 2)(x- 2)
We can see that a= 2, p= 2, and q= 2. Therefore, the point (2,0) is the only one x-intercept. It means that this point is also the of the given function.
Find and Graph the Axis of Symmetry
Since the parabola has only one x-intercept, which occurs at the point (2,0), the axis of symmetry is the x=2.
Identifying the y-intercept and its Reflection
We have found that in the given function the x-intercept is also the vertex, so we need to find other points to draw the parabola. We need at least three points. Recall the standard form in which the function was given.
f(x)=2x^2-8x+8
The y-intercept of the graph of a quadratic function written in standard form is given by the value of c. Thus, the point where our graph intercepts the y-axis is (0,8). Let's plot this point and its reflection across the axis of symmetry.
Draw the Parabola
Finally, we will draw the parabola through the three points we have.
We can see above that there are no restrictions on the x-variable. Furthermore, the y-variable takes values greater than or equal to 0. We can write the and of the function using this information.
Domain:& All real numbers
Range:& y ≥ 0
