# Find Quadratic Functions given their graphs

Find a quadratic function given its Graph. Examples with detailed solutions are presented. A tutorial with examples on graph of quadratic functions might help in understanding the present examples on finding quadratic equations.

## Review

A quadratic function f in vertex form is written asf(x) = a(x - h)

^{2}+ k

where h and k are the x and y coordinates respectively of the vertex (minimum or maximum) point of the graph.

The graph of of f is a parabola with the vertical line x = h as an axis of symmetry.

## Find quadratic function knowing its vertex and a point

### Example 1

Find the quadratic function f whose graph is shown below.### Solution to Example 1

Let h and k be the coordinates of the the vertex of the graph of function f. From the graph, the vertex (minimum point) is identified as (h , k) = (0 , 2) hence the vertex form of function f may be written asf(x) = a(x - h)

^{2}+ k = a(x - 0)

^{2}+ 2 = a x

^{2}+ 2

The point (1,3) on the graph of f will now be used to find coefficient a.

f(1) = a (1)

^{2}+ 2 = 3

Solve the above for a to obtain

a = 1

Hence

f(x) = x

^{2}+ 2

### Example 2

Find the quadratic function g whose graph is shown below and evaluate g(-3).### Solution to Example 2

The vertex of the graph of function g is a maximum point located at (h , k) = (0 , -1). Hence function g in vertex form is written asg(x) = a(x - h)

^{2}+ k = a(x - 0)

^{2}- 1 = a x

^{2}- 1

Coefficient a will now be found using the point (1,-2) that is on the graph of g.

g(1) = a (1)

^{2}- 1 = - 2

Solve the above for a to obtain

a = - 1

Hence g(x) is given by

g(x) = - x

^{2}- 1

g(- 3) = - (- 3)

^{2}- 1 = - 10

### Example 3

Find the quadratic function l whose graph is shown below and calculate the x-intercepts of the graph.### Solution to Example 3

The graph of function l has a vertex (maximum point) located at (h , k) = (2 , 1). Function l in vertex form is written asl(x) = a(x - h)

^{2}+ k = a(x - 2)

^{2}+ 1

We use the y intercept (0,-7) of the graph of l to find coefficient a as follows.

l(0) = a(0 - 2)

^{2}+ 1 = - 7

Solve the above for a to obtain

a = - 2

Function l(x) is given by

l(x) = - 2(x - 2)

^{2}+ 1

We now calculate the x intercepts by solving the equation

- 2(x - 2)

^{2}+ 1 = 0

2(x - 2)

^{2}= 1

Extract the square root to obtain the 2 solutions

x = 2 - √(1/2) and x = 2 + √(1/2)

and therefore the x intercepts are located at the points

(2 - √(1/2) , 0) and (2 + √(1/2) , 0)

## Find quadratic function knowing its x and y intercepts

### Example 4

Find the quadratic function s in standard form whose graph is shown below.### Solution to Example 4

The graph of function s has two x intercepts: (-1 , 0) and (2 , 0) which means that the equation s(x) = 0 has two solutions x = - 1 and x = 2. Hence s(x) can be written as the product of two factors as follows:s(x) = a(x + 1)(x - 2)

We now use the y intercept (0,- 4) of the graph of k to find coefficient a as follows.

s(0) = a(0 + 1)(0 - 2) = - 4

Solve the above equation for a to obtain

a = 2

Function s(x) is given by

s(x) = 2(x + 1)(x - 2)

We expand and simplify to write s(x) in standard form.

s(x) = 2 x

^{2}- 2 x - 4

## Find quadratic function knowing its axis and two points

### Example 5

Find the quadratic function m in standard form whose graph is a parabola with an axis of symmetry given by the vertical line x = -3 as shown below.### Solution to Example 4

The graph has an axis of symmetry given by the vertical line x = - 3 hence the x coordinate h of the vertex is equal to - 3 and m(x) may be written asm(x) = a(x + 3)

^{2}+ k

We now have two unknown a and k to determine. We use the points (-5 , 0) and (-2 , -3/2) shown on the graph of m to write two equations in a and k

The point (-5 , 0) means m(-5) = 0 which gives the equation a(- 5 + 3)

^{2}+ k = 0

The (-2 , -3/2) means m(-2) = -3/2 which gives the equation a(- 2 + 3)

^{2}+ k = -3/2

Simplify to obtain the system of equations

4 a + k = 0

a + k = - 3/2

Solve the system to obtain

a = 1/2 and k = -2

m(x) = (1/2)(x + 3)

^{2}- 2

Expand and rewrite m(x) in standard form

m(x) = (1/2)x

^{2}+ 3x + 5/2

## Find quadratic function knowing three points

### Example 6

Find the quadratic function w in standard form whose graph is a parabola shown below.### Solution to Example 6

The quadratic function w(x) in standard form is written as followsw(x) = a x

^{2}+ b x + c

We need to find the coefficients a, b and c. We use the three points on the graph of w to write 3 equations in a, b and c as follows:

point (0,-1/6) gives the equation: w(0) = a (0)

^{2}+ b (0) + c = - 1/6 (eq 1)

point (1 , 0) gives the equation: w(1) = a (1)

^{2}+ b (1) + c = 0 (eq 2)

point (3 , 10/3) gives the equation: w(3) = a (3)

^{2}+ b (3) + c = 10/3 (eq 3)

eq 1 simplifies to

c = - 1/6

Substitute c by - 1/6 in eq 2 and 3 to obtain two equations in a and b

a + b = 1/6

9a + 3 b = 7/2

Solve the above system of equations to obtain

a = 1/2 , b = -1/3

Substitute a, b and c by their values to write the quadratic function w(x) in standard form as follows

w(x) = (1/2) x

^{2}- (1/3) x - 1/6

## More References and links to quadratic functions and parabolas

GraphVertex and Intercepts Parabola Problems.

Find the Points of Intersection of a Parabola with a Line.

Parabola Problem with Solution.

Find the Points of Intersection of a Parabola with a Line.

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