How to Graph Quadratic Functions?

Graphing quadratic functions in vertex form

Here, the vertex of the parabola is \((h, k) = (-\frac{b}{2a}, -\frac{D}{4a})\). Now, to plot the graph of \(f(x)\), we start by taking the graph of \(x^2\), and applying a series of transformations to it:

Step 1: \(x^2\) to \(ax^2\): This means the vertical scale of the original parabola. If \(a\) is negative, the parabola will also flip its mouth from the positive to the negative side. The magnitude of the scaling depends upon the magnitude of \(a\).

Step 2: \(ax^2\) to \(a(x + \frac{b}{2a})^2\): This is a horizontal shift of magnitude \(|\frac{b}{2a}|\) units. The direction of the shift will be decided by the sign of \(\frac{b}{2a}\). The new vertex of the parabola will be at \((-\frac{b}{2a},0)\). The following figure shows an example shift:

Step 3: \(a(x + \frac{b}{2a})^2\) to \(a(x + \frac{b}{2a})^2 – \frac{D}{4a}\): This transformation is a vertical shift of magnitude \( |\frac{D}{4a}|\) units. The direction of the shift will be decided by the sign of \(\frac{D}{4a}\). The final vertex of the parabola will be at \((-\frac{b}{2a}, -\frac{D}{4a})\). The following figure shows an example shift:

Graphing quadratic functions in standard form

The general equation of a quadratic function is \(f(x) = ax^2+bx + c\). To plot the quadratic functions using the standard form of the function, we can convert the general form to the vertex form and then plot the quadratic function diagram or determine the axis of symmetry and y-intercept of the graph and plot it.

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Exercises for Graphing Quadratic Functions

Sketch the graph of each function.

• \(\color{blue}{f\left(x\right)=1-2x-3x^2}\)

• \(\color{blue}{f\left(x\right)=x^2+5x-4}\)

\(\color{blue}{f\left(x\right)=1-2x-3x^2}\)

• \(\color{blue}{f\left(x\right)=x^2+5x-4}\)

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