Vertex and x-y intercept of quadratic?

Question

A quadratic function is given. 
(a) Express the quadratic function in standard form 
(b) Find its vertex and its x- and y-intercept(s) 
(c) Sketch its graph 
f(x) = 2x^2 + 6x 
f(x) = 2x^2 + 4x + 3 
Thank you

Answer 1

(2)

f(x) = 2x² + 4x +3

(a) The quadratic function in standard form  is ax²+bx+c = 0

The given function is already in standard form i.e f(x) = 2x² + 4x +3

(b) y = 2x² + 4x +3

Convert the equation into vertex form of parabola is y  = a(x - h) ² + k

y = 2(x² + 2x + (1)² ) +3 -2

y = 2(x² + 2x + (1)² ) +1

y = 2(x + 1)² +1

Compare it to vertex form of parabola is y  = a(x - h) ² + k

where vertex of parabola is (h, k) = (-1 , 1)

 

To find y intercept substitute x = 0 in the equation y = 2x² + 4x +3 ⇒ y = 3

y intercept is y = 3

 

To find x intercept substitute y = 0 in the equation y = 2x² + 4x +3 

2x² + 4x +3 = 0

Since x is not real there is no x intercept 

 

(c) 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 = 2x² + 4x +3	(x, y)
0	y = 2(0)² +4 (0) + 3 = 3	(0, 3)
- 1	y =2(-1)²+4 (-1)+ 3  =  1	(-1, 1)
-2	y = 2(-2)2 +4(-2)+3 = 3	(-2 , 3)
0.5	y =2(0.5)2+4(0.5)+ 3 = 5.5	(0.5, -1.25)
-2.5	y= 2(-2.5)2+4(-2.5)+3 = 5.5	(-2.5, 5.5)
-1.5	y= 2(-1.5)2+4(-1.5)+3 = 5.5	(-1.5,1.5)

1.Draw a coordinate plane.

2.Plot the coordinate points.

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

Answer 2

(1)

f(x) = 2x² + 6x 

(a) The quadratic function in standard form  is ax²+bx+c = 0

The given function is already in standard form i.e f(x) = 2x² + 6x 

(b) y = 2x² + 6x 

Convert the equation into vertex form of parabola is y  = a(x - h) ² + k

y = 2(x² + 3x + (3/2)² ) - 2*(3/2)²

y = 2(x² + 3x + (3/2)² ) - 9/2

y = 2(x + 3/2)² - 9/2

Compare it to vertex form of parabola is y  = a(x - h) ² + k

where vertex of parabola is (h, k) = (-3/2 , -9/2)

 

To find y intercept substitute x = 0 in the equation y = 2x² + 6x ⇒ y = 0

y intercept is y = 0

 

To find x intercept substitute y = 0 in the equation y = 2x² + 6x 

2x² + 6x  = 0

x(2x+6) = 0

x = 0 or 2x+6 = 0

x = 0 or 2x = -6

x = 0 or x = -3

x intercept is x = 0 , -3

(c)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	2x² + 6x	(x, y)
0.5	2(0.5)²+6(0.5) =3.5	(0.5,3.5)
0	2(0)²+6(0) =0	(0,0)
-1	2(-1)²+6(-1) =-4	(-1,-4)
-1.5	2(-1.5)²+6(-1.5) =-4.5	(-1.5,4.5)
-3	2(-3)²+6(-3) =0	(-3,0)
-3.5	2(-3.5)²+6(-3.5) = 3.5	(-3.5,3.5)

1.Draw a coordinate plane.

2.Plot the coordinate points.

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