We are given that the x-intercepts occur at (2,0) and (7,0). Let's plot these points.
Find and Graph the Axis of Symmetry
The axis of symmetry is halfway between x-intercepts (x_1,0) and (x_2,0). Since we know that x_1=2 and x_2=7, the axis of symmetry of our parabola is halfway between (2,0) and (7,0).
x=x_1+ x_2/2 ⇒ x=2+ 7/2=9/2=4.5
We found that the axis of symmetry is the vertical line x=4.5.
Graph the y-intercept and its Reflection
We are given that the y-intercept 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 points we found.
b We want to draw a graph of the parabola with the given y-intercept and exactly one x-intercept. We will follow three steps to graph the function.
Graph the y-intercept and its reflection across the axis of symmetry.
Connect the points with a parabola.
Let's go through these steps one at a time.
Identify the Vertex and the Axis of Symmetry
When a quadratic function has only one x-intercept, it means that this point is also a vertex of the parabola. Since our function has the x-intercept at (- 1,0), we know that the parabola has the vertex at the point (- 1,0).
The axis of symmetry is a vertical line that divides the graph into two mirror images. Since the axis of symmetry of the parabola intersects vertex, we know that the equation of the axis is equal to the x-coordinate of the vertex. In our case it is x=- 1.
Graph the y-intercept and its Reflection
We are given that the y-intercept is (0,3). Let's plot this point and its reflection across the axis of symmetry.
Draw the Parabola
We can now draw the graph of the function. Let's connect the three points with a smooth curve.
c To draw the graph of the given function, we will follow four steps.
Identify and plot the x-intercepts.
Find and graph the axis of symmetry.
Find and plot the vertex.
Draw the parabola through the vertex and the points where the x-intercepts occur.
Let's go through these steps one at a time.
Identify and Plot the x-intercepts
The function is given in the factored form of a quadratic function.
y=a(x+b)(x+c)
In this form, where a ≠0, the x-intercepts are (- b,0) and (- c,0). Let's consider the factored form of our function.
y=(x+5)(x-1)
⇕
y= 1(x+ 5)(x+( - 1))
We can see that a= 1, b= 5, and c= - 1. Therefore, the x-intercepts occur at ( - 5,0) and ( 1,0).
Find and Graph the Axis of Symmetry
The axis of symmetry is halfway between the x-intercepts (- b,0) and (- c,0). Since we know that - b=- 5 and - c=1, the axis of symmetry of our parabola is halfway between (- 5,0) and (1,0).
x=- b+( - c)/2 ⇒ x=- 5+ 1/2=- 4/2=- 2
We found that the axis of symmetry is the vertical line x=- 2.
Find and Plot the Vertex
Since the vertex lies on the axis of symmetry, its x-coordinate is - 2. To find the y-coordinate, we will substitute - 2 for x in the given equation.
