The related graph of the given quadratic equation is the graph of the quadratic function on the left-hand side of the equation.
Equation:& x^2-4x+1=0
Related Function:& y=x^2-4x+1
To draw the graph of the quadratic function, written in standard form, we must start by identifying the values of a, b, and c.
y=x^2-4x+1 ⇔ y=1x^2+(- 4)x+1
We can see that a=1, b=- 4, and c=1. Now, we will follow four steps to graph the function.
The axis of symmetry of the parabola is the vertical line with equation x=2.
Calculating the Vertex
To calculate the vertex, we need to think of y as a function of x, y=f(x). We can write the expression for the vertex by stating the x- and y-coordinates in terms of a and b.
Vertex: ( - b/2a, f( - b/2a ) )
Note that the formula for the x-coordinate is the same as the formula for the axis of symmetry, which is x=2. Thus, the x-coordinate of the vertex is also 2. To find the y-coordinate, we need to substitute 2 for x in the function's rule.
We found the y-coordinate, and now we know that the vertex is (2,- 3).
Identifying the y-intercept and its Reflection
The y-intercept of the graph of a quadratic function written in standard form is given by the value of c. Thus, the point where our graph intercepts the y-axis is (0,1). Let's plot this point and its reflection across the axis of symmetry.
Connecting the Points
We can now draw the graph of the function. Since a=1, which is positive, the parabola will open upwards. Let's connect the three points with a smooth curve.
The roots are the x-coordinates where graph crosses the x-axis. By looking at the graph, we can state that the roots are located between 0 and 1, and between 3 and 4.
