x-intercepts: 2 and - 6 Increasing Interval: To the right of x=- 2 Decreasing Interval: To the left of x=- 2 Graph:
Practice makes perfect
We will find the intercepts and the increasing and decreasing intervals of the given quadratic function. Then we will draw the graph to verify our answer.
x-intercepts
Let's rewrite the function to match the intercept form. Then we can identify the x-intercepts, p and q.
f(x)=1/2 ( x-2 ) ( x+6 )
⇕
f(x)= 1/2 ( x- 2 ) ( x-( - 6) )
The x-intercepts of the given function are p= 2 and q= - 6.
Increasing and Decreasing Intervals
To describe where the graph is increasing and decreasing, we need to find the axis of symmetry of the parabola.
Axis of Symmetry: x = p+q/2
We can find this by substituting 2 and - 6 for p and q in the above formula.
The axis of symmetry is the vertical line x=- 2. Since a= 12 is greater than 0, the parabola opens upwards. Thus, the curve decreases to the left of x=- 2 and increases to the right of x=- 2.
Graph
We already know the x-intercepts of the parabola. Therefore, to draw the graph we only need to find the vertex and join the three point with a smooth curve. Since the axis of symmetry is the line x=- 2, the x-coordinate of the vertex is - 2. To find its y-coordinate, we will substitute - 2 for x in the given equation.
