The solutions, or roots, of ax^2+bx+c=0 are the x-intercepts of the graph of f(x)=ax^2+bx+c. Notice that our equation already is in standard form. Now we can identify the function related to the equation.
Equation:& - x^2+3x-1=0
Related Function:& f(x)=- x^2+3x-1
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 + 3x - 1 ⇕ f(x)=( -1)x^2 + 3x + ( -1)
We can see that a= -1, b= 3, and c= -1. Now, we will follow three steps to graph the function.
The axis of symmetry of the parabola is the vertical line with equation x=1.5.
Making the Table of Values
Next, we will make a table of values using x values around the axis of symmetry x=1.5.
x
-x^2+3x-1
f(x)
0
-( 0)^2+3( 0)-1
-1
0.5
-( 0.5)^2+3( 0.5)-1
0.25
1.5
-( 1.5)^2+3( 1.5)-1
1.25
2.5
-( 2.5)^2+3( 2.5)-1
0.25
3
-( 3)^2+3( 3)-1
-1
Plotting and Connecting the Points
We can finally draw the graph of the function. Since a= -1, which is negative, the parabola will open downwards. 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 0.4 and 2.6.
