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.

f(x) = a(x - h)

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.

f(x) = a(x - h)

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

f(1) = a (1)

Solve the above for a to obtain

a = 1

Hence

f(x) = x

g(x) = a(x - h)

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

g(1) = a (1)

Solve the above for a to obtain

a = - 1

Hence g(x) is given by

g(x) = - x

g(- 3) = - (- 3)

l(x) = a(x - h)

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

l(0) = a(0 - 2)

Solve the above for a to obtain

a = - 2

Function l(x) is given by

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

We now calculate the x intercepts by solving the equation

- 2(x - 2)

2(x - 2)

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)

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

m(x) = a(x + 3)

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)

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

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)

Expand and rewrite m(x) in standard form

m(x) = (1/2)x

w(x) = a x

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)

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

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

Vertex 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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