If a quadratic function is written in intercept form f(x)=a(x-p)(x-q), the x-intercepts are p and q.
x-intercepts: - 1 and 3 Increasing Interval: To the right of x=1 Decreasing Interval: To the left of x=1 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.
y=3/4 ( x+1 ) ( x-3 )
⇕
y= 3/4 ( x-( - 1) ) ( x- 3 )
The x-intercepts of the given function are p= - 1 and q= 3.
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 - 1 and 3 for p and q in the above formula.
The axis of symmetry is the vertical line x=1. Since a= 34 is greater than 0, the parabola opens upwards. Thus, the curve decreases to the left of x=1 and increases to the right of x=1.
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=1, the x-coordinate of the vertex is 1. To find its y-coordinate, we will substitute 1 for x in the given equation.
