Skip to content# Sketching Quadratic Graphs Step-by-Step Examples

## What is Sketching Quadratic Graphs?

## Key Points on a Quadratic Graph

## How to Sketch a Quadratic Graph

## Conclusion

### Practice Questions: Sketching Quadratic Graphs

## Solutions:

**Sketching Quadratic Graphs**

- Quadratic functions are an important part of mathematics, and graphing them is a crucial thing every learner or anyone involved in using mathematics should be in a position to do.
- Similar to parabola, Quadratic graphs show the pattern of the quadratic equation and present much more information about the function.

In this article, we will discuss:

**What is Sketching Quadratic Graphs?****Key Points on a Quadratic Graph****How to Sketch a Quadratic Graph**

Here is one more link to practice a few extra questions: Maths Genie Sketching Quadratic Graphs Questions

- A quadratic graph is the geometric representation of a quadratic equation which can also be written as
**f(x) = ax**where^{2}+ bx + c**a, b and c**are constants.

- Quadratic graphs are graphs of quadratic functions that resemble the letter ‘U’ and can either be a ‘U’ shape that opens up or down depending on the sign of
**‘a’.** - These are usual properties and characteristics of any quadratic graph, including vertex, symmetry axis, and intercepts.

**Vertex:**

- The vertex is the coordinate point for the graph which is the lowest point in the case of an up turned parabola or the highest point in the case of a down turned parabola.
- In the quadratic function of the form:

**f(x) = ax ^{2} + bx + c,**

the x-coordinate of the vertex is given by the formula

**x = -b/(2a).**

- The other value of the y-axis can then be determined by putting in the value of the x-coordinate into the original equation.

**Axis of Symmetry:**

- The axis of symmetry is a down-line passing through the vertex, which as well bisects the parabola into two equal halves.
- The equation of the axis of symmetry is
**x = -b/(2a)**

**Intercepts:**

- For quadratic graphs, the x-intercepts or intercepts points are the points at which they cross the x- axis. These are the values of x for which
**f(x) = 0.** - The y-intercept, denoted by the point
**(0,c)**. it is the point where the graph of a function line crosses the y axis.

**Direction of Opening:**

- Quadratic function shows the direction of the opening according to the coefficient
**‘a.’**This indicates that if variable**‘a’**is positive the parabola face will be upward bending whereas if the variable**‘a’**is negative the parabola face will be downward bending.

**Identify Key Parameters:**

- Let’s find out what values corresponds
**‘a,’****‘b,**’ and**‘c’**from the following quadratic function. - These are values that you can use to determine vertex, axis of symmetry and intercepts.

**Plot the Vertex:**

- To find the x-coordinate of the vertex, we need to use the formula
**x = -b/(2a).** - Plug in this value into the original equation to get the y-coordinate that corresponds to it.
- Locate the vertex from the graph.

**Draw the Axis of Symmetry:**

- The axis of symmetry can be found with the help of the given equation,
**x = (-b/2a).** - Place a vertical line at the vertex that will show the parabola’s axis of symmetry.

**Plot Intercepts:**

- Locate the x-intercept(s) by putting f(x) = 0 and solving for x.
- The y-intercept is the value of y when x = 0.
- Plot these points on the graph.

**Sketch the Parabola:**

- Join the plotted points so as to form a smooth curve that would be that of a parabola.
- Note the sign of
**‘a’**in the opening that dictates a change in direction.

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**Solved Example: **

**Question 1: Sketch the graph of the quadratic function f(x) = 2x ^{2} – 4x – 6.**

Solution:

**Step #1:**Identify Key Parameters:

**a = 2, b = -4, c = -6.**

**Step #2:**Plot the Vertex

Use the formula

**x = -b/(2a) **

**= -(-4)/(2 x 2) **

**= 1.**

Substitute x = 1 into f(x) to find y:

**f(1) = 2(1) ^{2} – 4(1) – 6 **

**= -8.**

Vertex coordinates: **(1, -8).**

**Step #3:**Draw the Axis of Symmetry:

The axis of symmetry is **x = 1.**

**Step #4:**Plot Intercepts:

x-intercepts:

**Set** **f(x) = 0 → 2x ^{2} – 4x – 6 = 0.**

Solve using factoring or quadratic formulas to find x-values.

y-intercept:

**Set x = 0 → f(0) = -6.**

**Step #5:****Sketch the Parabola:**

Connect the vertex and intercepts to form the parabola. Since **‘a’** is positive, the parabola opens upwards.

- Drawing quadratic graphs could be used as a graphical method of analyzing the behaviour of quadratic functions.
- That is the reason why it is necessary to find the vertex, axis of symmetry, intercepts, and the direction in which the parabola opens.
- Quadratic graphs are graphical representations of quadratic functions that can help better comprehend their behaviour.
- The skill of sketching quadratic graphs is relevant and can be used in solving problems in physics, engineering, economy and many other sciences.

**Question 1: **Sketch the following graphs y = x^{2} + 6x + 8

**Question 2:** Sketch the following graphs y = x^{2} – x – 6

**Question 3:** Sketch the following graphs y = x^{2} + 6x + 9

**Question 4:** Sketch the following graphs y = x^{2} – 13x + 42

**Question 5:** Sketch the following graphs y = x^{2} + 5x – 36

**Question 6:** Sketch the following graphs y = x^{2} – 2x + 1

**Question 7:** Sketch the following graphs y = x^{2} + 5x + 11

**Question 8:** Sketch the following graphs y = x^{2} – 4x + 7

**Question 9:** Sketch the following graphs y = x^{2} + 4x

**Question 10:** Sketch the following graphs y = x^{2} + 2x – 8

**Question 1:** Sketch the following graphs y = x^{2} + 6x + 8

Solution:

**Question 2:** Sketch the following graphs y = x^{2} – x – 6

Solution:

**Question 3:** Sketch the following graphs y = x^{2} + 6x + 9

Solution:

**Question 4:** Sketch the following graphs y = x^{2} – 13x + 42

Solution:

**Question 5:** Sketch the following graphs y = x^{2} + 5x – 36

Solution:

**Question 6:** Sketch the following graphs y = x^{2} – 2x + 1

Solution:

**Question 7:** Sketch the following graphs y = x^{2} + 5x + 11

Solution:

**Question 8:** Sketch the following graphs y = x^{2} – 4x + 7

Solution:

**Question 9:** Sketch the following graphs y = x^{2} + 4x

Solution:

**Question 10:** Sketch the following graphs y = x^{2} + 2x – 8

Solution: