The solutions, or roots, of ax^2+bx+c=0 are the x-intercepts of the graph of f(x)=ax^2+bx+c. Let's write our equation in standard form. This means gathering all of the terms on the left-hand side of the equation.
x^2=12 ⇔ x^2-12=0
Now we can identify the function related to the equation.
Equation:& x^2-12=0
Related Function:& f(x)=x^2-12
Graphing the Related Function
To draw the graph of the related function written in standard form, we must start by identifying the values of a, b, and c.
f(x)=x^2-12 ⇕ f(x)=( 1)x^2+( 0)x+( -12)
We can see that a= 1, b= 0, and c= -12. Now, we will follow three steps to graph the function.
The axis of symmetry of the parabola is the vertical line with equation x=0.
Making the Table of Values
Next, we will make a table of values using x values around the axis of symmetry x=0.
x
x^2-12
f(x)
-4
( -4)^2-12
4
-3
( -3)^2-12
-3
0
( 0)^2-12
-12
3
( 3)^2-12
-3
4
( 4)^2-12
4
Plotting and Connecting the Points
We can finally draw the graph of the function. Since a= 1, which is positive, the parabola will open upwards. Let's connect the points with a smooth curve.
Finding the x-intercepts
Let's identify the x-intercepts of the graph of the related function.
By looking at the graph, we can approximate values for the x-intercepts. We can see that the parabola intercepts the x-axis at -3.5 and 3.5.
