Draw the parabola through the vertex and the x-intercepts.
Let's do it!
Use the Quadratic Formula
In order to find the x-intercepts of the given function, we will substitute 0 for y and look for solutions to the resulting quadratic equation.
0 = 2x^2-x-15
Let's use the quadratic formula.
0= ax^2+ bx+ c ⇕ x=- b± sqrt(b^2-4 a c)/2 a
We first need to identify the values of a, b, and c.
0 = 2x^2-x-15 ⇕ 0= 2x^2+( - 1)x+( - 15)
We see that a= 2, b= - 1, and c= - 15. Let's substitute these values into the quadratic formula.
The solutions for this equation are x= 1± 114. Let's separate them into the positive and negative cases.
x=1± 11/4
x_1=1+11/4
x_2=1-11/4
x_1=12/4
x_2=-10/4
x_1=3
x_2=-2.5
We found that the solutions of the given equation are x_1= 3 and x_2= - 2.5. This means that the x-intercepts occur at ( 3,0) and ( -2.5,0).
Find and Graph the Axis of Symmetry
The axis of symmetry is halfway between (x_1,0) and (x_2,0). Since we know that x_2=- 2.5 and x_1=3, the axis of symmetry of our parabola is halfway between (- 2.5,0) and (3,0).
x=x_2+x_1/2 ⇒ x=- 2.5+ 3/2=0.25
We found that the axis of symmetry is the vertical line x=0.25.
Find and Plot the Vertex
Since the vertex lies on the axis of symmetry, its x-coordinate is 0.25. We can find the y-coordinate of the vertex if we substitute 0.25 for x in the given equation. Let's do it!
The y-coordinate of the vertex is - 15.125, so the vertex is the point (0.25,- 15.125).
Draw the Parabola
Finally, we let's draw the parabola through the vertex and the x-intercepts.
b Notice that the given function is the original one multiplied by -1. This changes the outputs of the original function to their opposites. Ultimately, this means that the graph will be a parabola with the same x-intercepts and axis of symmetry, but it will open downward instead of upward.
