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: - 5 and - 1 Increasing Interval: To the left of x=- 3 Decreasing Interval: To the right of x=- 3 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.
h(x)=- 5 ( x+5 ) ( x+1 )
⇕
h(x)= - 5 ( x-( - 5) ) ( x-( - 1) )
The x-intercepts of the given function are p= - 5 and q= - 1.
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 - 5 and - 1 for p and q in the above formula.
The axis of symmetry is the vertical line x=- 3. Since a= - 5 is less than 0, the parabola opens downwards. Thus, the curve increases to the left of x=- 3 and decreases to the right of x=- 3.
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 points with a smooth curve. Since the axis of symmetry is the line x=- 3, the x-coordinate of the vertex is - 3. To find its y-coordinate, we will substitute - 3 for x in the given equation.
