Draw the parabola through the vertex and the points where the x-intercepts occur.
Let's go through these steps one at a time.
Find and Plot the x-intercepts
To find the x-intercepts, think of the points where the graph of an equation crosses the x-axis. The y-value of that ( x, y) coordinate pairs is 0, and the x-values are the x-intercepts. To find the x-intercepts of the equation we should substitute 0 for y and solve for x.
0=3x^2-10x^2+2To solve the above quadratic equation for x we will 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=3x^2-10x^2+2 ⇕ 0= 3x^2+( - 10)x+ 2
We see that a= 3, b= - 10, and c= 2. Let's substitute these values into the Quadratic Formula.
Using the Quadratic Formula, we found that the solutions of the equation are x_1= 5-sqrt(19)3 ≈ 0.21 and x_2= 5+sqrt(19)3 ≈ 3.12.
Therefore, the x-intercepts of the parabola occur approximately at ( 0.21,0) and ( 3.12,0).
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_1 = 5-sqrt(19)3 and x_2 = 5+sqrt(19)3, the axis of symmetry of our parabola is halfway between ( 5-sqrt(19)3,0) and ( 5+sqrt(19)3,0).
x=x_1+x_2/2
⇓
x=5-sqrt(19)3+ 5+sqrt(19)3/2=103/2=5/3
We found that the axis of symmetry is the vertical line x= 53.
Find and Plot the Vertex
Since the vertex lies on the axis of symmetry, its x-coordinate is 53. To find the y-coordinate, we will substitute 53 for x in the given equation.
