graph y=(x-7)(x+3)

Question

confused please help

Answer 1

Given equation y = (x-7)(x+3)

y = x^2+3x-7x-21

y = x^2-4x-21

Compare it standard form of parabola y = ax^2+bx+c

a = 1, b = -4, c = -21

x = -b/2a = 4/2 = 2

substitute x = 2 in y = x^2-4x-21

y = 4-8-21 = -25

Vertex of parabola is (2,-25)

Graph of parabola

Answer 2

The standard form of parabola equation is y = a(x - h)^2 + k, where (h, k) = vertex and axis of symmetry x = h.

The equation is y = (x - 7)(x + 3).

Rewrite the equation as y = x² - 4x - 21.

The above equation represents a parabola.

Write the equation in standard form of a parabola equation.

To change the expression [x² - 4x - 21] into a perfect square trinomial add and subtract (half the x coefficient)²

 Here x coefficient = - 4. so, (half the x coefficient)² = (- 4/2)²= 4.

y = x² - 4x + 4 - 21 - 4

y = (x - 2)² - 25.

Compare the equation y = (x - 2)² - 25 with the standard form of parabola equation is y = a(x - h)^2 + k, where (h, k) = vertex and axis of symmetry x = h.

Vertex (h, k) = (2, - 25), and axis of symmetry x = 2.

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

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

x	y = (x - 2)2 - 25	(x, y)
- 4	y = (- 4 - 2)2 - 25 = (- 6)2 - 25 = 36 - 25 = 11	(- 4, 11)
- 3	y = (- 3 - 2)2 - 25 = (- 5)2 - 25 = 25 - 25 = 0	(- 3, 0)
- 2	y = (- 2 - 2)2 - 25 = (- 4)2 - 25 = 16 - 25 = - 9	(- 2, - 9)
- 1	y = (- 1 - 2)2 - 25 = (- 3)2 - 25 = 9 - 25 = - 16	(- 1, - 16)
0	y = (0 - 2)2 - 25 = (- 2)2 - 25 = 4 - 25 = - 21	(0, - 21)
2	y = (2 - 2)2 - 25 = (0)2 - 25 = - 25	(2, - 25)
4	y = (4 - 2)2 - 25 = (2)2 - 25 = 4 - 25 = - 21	(4, - 21)
6	y = (6 - 2)2 - 25 = (4)2 - 25 = 16 - 25 = - 9	(6, - 9)
8	y = (8 - 2)2 - 25 = (6)2 - 25 = 36 - 25 = 11	(8, 11)

 

Answer 3

Contd........

1.Draw a coordinate plane.

2.Plot the coordinate points.

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

Answer 4

Graph of the parabola in intercept form :

The intecept form of quadratic equation(parabola equation) is y = a(x - p)(x - q), where p and q are x - intercepts, and

• The role of ' a '.

1. If a > 0, then the parabola opens upwards.
2. If a < 0, then the parabola opens downwards.

The function is y = (x - 7)(x + 3).

Compare the equation with the intecept form of quadratic equation(parabola equation) is y = a(x - p)(x - q).

• a = 1 > 0 (positive), so the graph of function opens upwards and has a minimum value.
• x - intercepts p and q are 7 and - 3.
• y - intercept is a(- p)(- q) = 1(- 7)(3) = - 21.
• The y - intercept is (0, -21), so the point paired with it on other side of the axis of symmetry is (2, - 21) and has the same y - value.
  • Since, The axis of symmetry devides the parabola into two equal parts.So, if there is a point (0, - 21) on one side, there is a corresponding point on other side that is the same distance from the axis of symmetry and has the same y - value.
  • The distance between the points (0, - 21) and (2, - 21) = 2 = The distance between (2, - 21) and the point paired with it on other side of the axis of symmetry and has the same y - value.
  • The distance between (2, - 21) and the point paired with it on other side of the axis of symmetry. = 2 + 2 = 4.
  • Therefore, The point paired with it on other side of the axis of symmetry is (4, - 21). 
• The axis of symmetry is the line x = (p + q)/2.

Axis of symmetry x = (7 - 3)/2 = 4/2 = 2.

• The vertex (x, y) = (x, f(x))

= (2, f(2))

= (2, (2 - 7)(2 + 3) )

= (2, (- 5)(5)).

The vertex (x, y) = (2, - 25).

• Graph :

1.Draw a coordinate plane.

2.Plot the coordinate points.

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