The word transform means to change from one form to another. The following guide helps you learn how to transform quadratic equations.
Transformations of functions mean transforming the function from one form to another. Transformation of functions is a unique way of changing the formula of a function minimally and playing around with the graph.
Quadratic equations in standard form are represented as \(ax^2+bx+c=0\), where \(a≠0\) and \(a, b, c\) are real constants. The graph of a quadratic function is a parabola that can be represented in two forms:
The Vertex of a quadratic equation is its minimum or lowest point if the parabola is opening upwards or its the highest or maximum point if it opens downwards. In rearranging the standard form, the vertex is obtained by completing the square method. In comparing both forms,
When a quadratic function is represented as a vertex, the following points should be considered:
If \(k > 0\), the graph shifts upwards by \(k\) units.
To plot a quadratic function, the above steps are followed and subsequent transformations are used.
Consider the standard form of quadratic equation \(ax^2+bx+c=0\), having \(\alpha\) and \(\beta\) as roots. Suppose instead of \(\alpha\) and \(\beta\) as roots of the quadratic equation, roots are given as \((\alpha+m)\) and \((\beta+m)\). To draw such an equation, the main graph is shifted to the right, as shown in the figure below. The equation that shows this is as follows:
Note: If roots are \((\alpha-m)\) and \((\beta-m)\), the graph shifts left by \(m\) units.
Considering the vertex form i.e., \(f(x)=a(x-h)^2+k\), to shift the graph vertically changes are made in the value of \(k\).
Let’s first consider a graph \(y=x^2\), it represents a parabola with vertex at \((0,0)\) as shown:
If \(k>0\), the graph shifts upwards by \(k\) units:
Graph the function \(y=2x^2-5\).
Solution:
If we start with \(y=x^2\) and multiply the right side by \(2\), it stretches the graph vertically by a factor of \(2\).
Then if we subtract \(5\) from the right side of the equation, it shifts the graph down \(5\) units.
