## How to Graph Quadratic Functions

Read,5 minutes

### Graphing Quadratic Functions

Putting quadratic functions on a coordinate plane is the process of quadratic graphing functions.

This guide will teach you how to draw graphs of quadratic functions. Both **general** form and **vertex** form can be used to graph quadratic functions.

### How to draw graphs of quadratic functions, step by step

Graphing quadratic functions is a way to look at the **nature** of quadratic functions in a visual way. The **coefficient** \(a\) of the quadratic function \(f(x) \ = \ ax^2 \ + \ bx \ + \ c\), where \(a, \ b,\), and \(c\) are **real** numbers and \(a \ ≠ \ 0\), determines the **shape** of the parabola, which is the graph of a quadratic function.

A quadratic function's vertex form is \(f(x) \ = \ a(x \ - \ h)^2 \ +\ k\), where \((h,k)\) is the parabola's vertex. The **coefficient** of a quadratic function tells whether the graph goes **up** or **down**.

Note that the coefficient \(a\) also controls the **rate** at which the graph of the quadratic function goes up (or down) from the vertex. If \(a\) is **bigger** and positive, the process goes up more **quickly**, and the graph looks **thinner**.

### Using vertex form to draw graphs of quadratic functions

We will learn how to plot each quadratic function step by step. Think about the general quadratic function \(f(x) \ = \ ax^2 \ + \ bx \ + \ c\).

First, we rearrange it so that \(f(x) \ = \ ax^2 \ + \ bx \ + \ c\) becomes \(f(x) \ = \ a(x \ + \ \frac{b}{2a})^2 \ - \ \frac{Δ}{4a}\). The **discriminant** is the term \(Δ\), which is given by \(Δ \ = \ b^2 \ - \ 4ac\).

Here, the point where the parabola **starts** and **ends** is \((h \ , \ k) \ = \ (-\frac{b}{2a} \ , \ -\frac{Δ}{4a})\). Now, to draw the graph of \(f(x)\), we start with the graph of \(x^2\) and **change** it in several ways:

**Step 1:** Change \(x^2\) to \(ax^2\). This is the original parabola's vertical **scale**. If \(a\) is negative, the **mouth** of the parabola will also change from positive to negative. How much the scaling goes **up** or **down** depends on how **big** \(a\) is.

Consider this graph as \(x^2\):

If \(a \ > \ 0\), \(ax^2\) becomes like this:

But if \(a \ < \ 0\), \(ax^2\) becomes like this:

**Step 2: **Change \(ax^2\) to a(x \ + \ \frac{b}{2a})^2. This is a **horizontal shift** of \(|\frac{b}{2a}|\) units. The **sign** of \(\frac{b}{2a}\) will show which way the shift will go. The new point of the parabola's vertex will be \((\frac{b}{2a} \ , \ 0)\). The figure below shows an **example** of a shift:

If \(\frac{b}{2a} \ = \ -2\):

**Step 3:** Change \(a(x \ + \ \frac{b}{2a})^2\) to \(a(x \ + \ \frac{b}{2a})^2 \ - \ \frac{Δ}{4a}\) . This is a **vertical** shift by \(|\frac{Δ}{4a}|\) units. The **sign** of \(\frac{Δ}{4a}\) will show which way the shift will go. The point where the parabola **ends** will be at \((-\frac{b}{2a} \ , \ -\frac{Δ}{4a})\). The figure below shows an **example** of a shift:

If \(\frac{b}{2a} \ = \ -3 \ , \ \frac{Δ}{4a} \ = \ 4\):

### Using standard form to draw graphs of quadratic functions

\(f(x) \ = \ ax^2 \ + \ bx \ + \ c\) is the **general** equation for a quadratic function. Using the standard form of the function, we can **convert** the general form to the **vertex** form, then draw the quadratic function diagram, or find the axis of **symmetry** and **y-intercept** of the graph and draw it.

### Exercises for Graphing Quadratic Functions

**1) **Sketch the graph: \(x^2 \ - \ 3\)

**2) **Sketch the graph: \(x^2 \ + \ 1\)

**3) **Sketch the graph: \((x \ - \ 2)^2 \ + \ 1\)

**4) **Sketch the graph: \((x \ + \ 3)^2 \ - \ 3\)

**5) **Sketch the graph: \((x \ + \ 2.5)^2 \ - \ 4\)

**6) **Sketch the graph: \((x \ - \ 1)^2 \ - \ 5\)

**7) **Sketch the graph: \(x^2 \ + \ 4\)

**8) **Sketch the graph: \((x \ - \ 2)^2 \ + \ 3\)

**9) **Sketch the graph: \((x \ + \ 3)^2 \ + \ 4\)

**10) **Sketch the graph: \((x \ + \ 2)^2 \ - \ 5\)

**1) **Sketch the graph: \(x^2 \ - \ 3\)

**2) **Sketch the graph: \(x^2 \ + \ 1\)

**3) **Sketch the graph: \((x \ - \ 2)^2 \ + \ 1\)

**4) **Sketch the graph: \((x \ + \ 3)^2 \ - \ 3\)

**5) **Sketch the graph: \((x \ + \ 2.5)^2 \ - \ 4\)

**6) **Sketch the graph: \((x \ - \ 1)^2 \ - \ 5\)

**7) **Sketch the graph: \(x^2 \ + \ 4\)

**8) **Sketch the graph: \((x \ - \ 2)^2 \ + \ 3\)

**9) **Sketch the graph: \((x \ + \ 3)^2 \ + \ 4\)

**10) **Sketch the graph: \((x \ + \ 2)^2 \ - \ 5\)