how do you graph y=-2x^2-4x+3

Question

I need to graph parabolas in standard, vertex and intercept form.

Answer 1

Given equation y = -2x^2-4x+3

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

a = -2, b = -4, c = 3

Axs of symmetry x = -b/2a

x = -(-4)/2(-2)

x = -1

Substitute x value in given equation.

y = -2+4+3 = 5

Vertex of parabola = (-1,5)

To find y intercepts substitute x = 0 in given equation.

y = 3

To find x intercept substitute y = 0 in given equation.

0 = -2x^2-4x+3

Multple to each side by negitive one.

2x^2+4x-3 = 0

Compare it quadratic equation ax^2+bx+c = 0

a = 2, b = 4, c = -3

Roots are x = [-b±√b^2-4ac]/2a

x = [-4±√16+24]/4

x = [-4±√40]/4

x = [-4±6.32]/4

x = (-4+6.32)/4 = 2.32/4 = 0.58

x = (-4-6.32)/4 = -10.32/4  = -2.58

x intercepts are x = 0.58,-2.58

Graph of parabola

draw the coordinate plane.

Plot the vertex of parabola.

Draw the curve neatly through the vertex to satisify the condition of x,y intercepts.

then formed parabola  indicating given parabola.

Answer 2

Graph of the parabola in standard form :

The function is f(x) = y = - 2x² - 4x + 3.

The standard form of parabola equation is f(x) = ax² + bx + c.

Step 1 :

Find the axis of symmetry :

Formula for the equation of the axis of symmetry:  x = - b/2a.

Substitute the values of b = - 4 and a = - 2 in the formula, x = - b/2a.

x = - (- 4)/2(- 2)

x = 4/- 4 = - 1.

The equation for the axis of symmetry is x = - 1.

Step 2 :

Find the vertex :

To find the vertex, use the value of equation for the axis of symmetry as the  x - coordinate of the vertex.

To find the y - coordinate, substitute the value of x = - 1 in the original equation, y = - 2x² - 4x + 3.

y = - 2(- 1)² - 4(-1) + 3

y = - 2 + 4 + 3 = 7 - 2

y = 5.

The vertex is (- 1, 5).

Determine whether the function has maximum or minimum value :

The value of a = - 2 < 0 (negative), so the graph of function opens downward and has a maximum value.

The maximum value (y - coordinate of the vertex) is 5.

Step 3 :

Find the y -  intercept :

To find the y - intercept, Substitute the value x = 0 in the original function, f(x) = y = - 2x² - 4x + 3.

y = - 2(0)² - 4(0) + 3

y = 3.

The y - intercept is 3.

Step 4 :

• Lets find the another point.Choose an x - value of 1 and substitute.The new point is at (1, - 3).
• The point paired with it on other side of the axis of symmetry is (- 3, - 3) and has the same y - value.
• Since, The axis of symmetry devides the parabola into two equal parts.So, if there is a point (1, - 3) 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 (1, - 3) and (-1, - 3) = 2 = The distance between (- 1, - 3) and the point paired with it on other side of the axis of symmetry and has the same y - value.
• The distance between (- 1, - 3) and the point paired with it on other side of the axis of symmetry. = - 1 - 2 = - 3.
• Therefore, The point paired with it on other side of the axis of symmetry is (- 3, - 3).
• The y - intercept is (0, 3), so the point paired with it on other side of the axis of symmetry is (-1, 3) and has the same y - value.
• Therefore, The point paired with it on other side of the axis of symmetry is (- 2, 3).
• Connect these points and create a smooth curve.

Graph :

Answer 3

Contd.........

Graph of the parabola in vertex form :

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

The parabola is f(x) = y = - 2x² - 4x + 3.

Write the equation in vertex form of a parabola eqaution.

Divide each side by negative 2.

- y/2 = x² + 2x - 3/2

To change the expression [x² + 2x - 3/2] into a perfect square trinomial add and subtract (half the x coefficient)²

 Here x coefficient = 2. so, (half the x coefficient)² = (2/2)²= 1.

- y/2 = x² + 2x + 1 - 3/2 - 1

- y/2 = (x + 1)² - 5/2.

y = - 2(x + 1)² + 5.

Compare the equation y = - 2(x + 1)² + 5 with the vertex form of parabola equation is y = a(x - h)^2 + k, where (h, k) = vertex and axis of symmetry x = h.

Vertex (h, k) = (- 1, 5), and axis of symmetry x = - 1.

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 = - 2(x + 1)2 + 5	(x, y)
- 3	y = - 2(- 3 + 1)2 + 5 = - 2(- 2)2 + 5 = - 8 + 5 = - 3	(- 3, - 3)
- 2	y = - 2(- 2 + 1)2 + 5 = - 2(- 1)2 + 5 = - 2 + 5 = 3	(- 2, 3)
- 1	y = - 2(- 1 + 1)2 + 5 = - 2(0)2 + 5 = 0 + 5 = 5	(- 1, 5)
0	y = - 2(0 + 1)2 + 5 = - 2(1)2 + 5 = - 2 + 5 = 3	(0, 3)
1	y = - 2(1 + 1)2 + 5 = - 2(2)2 + 5 = - 8 + 5 = - 3	(1, - 3)

1.Draw a coordinate plane.

2.Plot the coordinate points.

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

Answer 4

Contd..........

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 f(x) = y = - 2x² - 4x + 3.

Write the parabolaequationin intercept form.

Factor the expression - 2x² - 4x + 3.

- 2x² - 4x + 3, is a quadratic.So, use quadratic formula to find the roots of the related quadratic equation.

x = [ - b ± √(b² - 4ac) ]/2a.

Substitute a = - 2, b = - 4, and c = 3.

x = [ 4 ± √(16 + 24) ]/2*-2

x = - [ 4 ± √40 ]/4

x = - [ 2 ± √10 ]/2.

From foil method : y = - 1(x - [- ( 2 - √10)/2])(x - [ (√10 +2)/2])

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

• a = - 1 > 0 (negative), so the graph of function opens downwards and has a maximum value.
• x - intercepts p and q are -( 2 - √10)/2 and (√10 +2)/2.
• y - intercept is a(- p)(- q) = - 1([( 2 - √10)/2])(- (√10 +2)/2) = 3.
• The y - intercept is (0, 3), so the point paired with it on other side of the axis of symmetry is (- 2, 3) 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, 3) 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, 3) and (- 1, 3) = 1 = The distance between (-1, 3) and the point paired with it on other side of the axis of symmetry and has the same y - value.
  • The distance between (- 1, 3) and the point paired with it on other side of the axis of symmetry. = - 1 - 1 = - 2.
  • Therefore, The point paired with it on other side of the axis of symmetry is (- 2, 3).
• The axis of symmetry is the line x = (p + q)/2.

Axis of symmetry x = (- ( 2 - √10)/2 + ((√10 +2)/2))/2 = - 4/4 = - 1.

• The vertex (x, y) = (x, f(x))

= (- 1, f(- 1))

= (- 1, 5).

The vertex (x, y) = (- 1, 5).

•  

Graph :

1.Draw a coordinate plane.

2.Plot the coordinate points.

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