Quadratic function in standard form, min and max?

Question

A quadratic function is given 
(a) Express the quadratic function in standard form 
(b) Sketch its graph 
(c) Find its maximum or minimum value 
f(x) = 3x^2 - 6x + 1 
f(x) = -x^2 - 3x + 3 

Answer 1

The quadratic function is a parabola.

The function f(x) = 3x² - 6x + 1

The standard form of parabola is y = ax² + bx + c.

a) Standard form is y = 3x² - 6x + 1

a = 3, b = - 6, c = 1.

b) Graph

Make the table of values to find ordered pairs that satisfy the equation.

Choose the  random values for x and find the corresponding values for y.

x	y = 3x2 - 6x + 1	(x, y)
0	y = 3(0)2- 6(0) + 1 = 1	(0, 1)
- 0.5	y = 3(-0.5)2 - 6(-0.5) + 1 =  4.75	(-0.5, 4.75)
1	y = 3(1)2 - 6(1) + 1 = -2	(1, -2)
0.5	y = 3(0.5)2 -6(0.5) + 1 = - 1.25	(0.5, -1.25)
2	y= 3(2)2-6(2)+1 = 1	(2, 1)
2.5	y=3(2.5)2-6(2.5)+1 = 4.75	(2.5, 4.75)

1.Draw a coordinate plane.

2.Plot the coordinate points.

3.Then sketch the graph, connecting the points with a smooth curve.

c) a = 3 is positive number the parabola opens up and it has minimum point which is the vertex of parabola.

Axis of symmetry x = - b/2a = - (- 6)/2(3) = 1.

Substitute value of x in y = 3x² - 6x + 1.

y = 3(1)² - 6(1) + 1

y = - 2

x, y are the coordinates of vertex.

Vertex is the minimum point of the parabola

Minimum value is (1, - 2).

Answer 2

The function f(x) = - x² - 3x + 3

The standard form of parabola is y = ax² + bx + c.

a) Standard form is y = - x² - 3x + 3

a = -1, b = - 3, c = 3.

b) Graph

Choose the  random values for x and find the corresponding values for y.

x	y = - x2 - 3x + 3	(x, y)
0	y= -(0)2-3(0)+3= 3	(0, 3)
- 2	y=-(-2)2-3(-2)+3=5	(-2, 5)
-3	y=-(-3)2-3(-3)+3=3	(-3,3)
-4	y=-(-4)2-3(-4)+3=- 1	(-4,-1)
1	y=-(1)2-3(1)+3= -1	(1, -1)

1.Draw a coordinate plane.

2.Plot the coordinate points .

3.Then sketch the graph, connecting the points with a smooth curve.

c)a = -1 is negative number the parabola opens down and it has maximum point which is the vertex of parabola.

Axis of symmetry x = - b/2a = - (- 3)/2(-1) = -1.5.

Substitute value of x in y = - x² - 3x + 3

y =-(-1.5)² - 3(-1.5) + 3  = 5.25

x, y are the coordinates of vertex.

Vertex is the maximum point of the parabola.

maximum value is (-1.5, 5.25).