Question

find the vertex & graph g(x)=2x^2-x-3

Answer 1

Given parabola g(x) = 2x^2-x-3

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

a = 2, b = -1, c = -3.

Axis of symmetry x = -b/2a  = 1/4

To find vertex substitute x value in given equation.

y = 2(1/4)^2-1/4-3

y = 2*1/16-1/4-3

y = 1/8-13/4

y = -25/8

Vertex of parabola = (1/4,-25/8) 

 

Answer 2

The function is g (x) = y = 2x² - x - 3.

The standard form of quadratic function 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 = - 1 and a = 2 in the formula, x = - b/2a.

x = - (- 1)/2(2)

x = 1/4.

The equation for the axis of symmetry is x = 1/4.

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 =1/4 in the original function, y = 2x² - x - 3.

y = 2(1/4)² - (1/4) - 3

y = 2(1/16) - 1/4 - 3

y = - 25/8.

The vertex is (1/4, - 25/8).

Determine whether the function has maximum or minimum value :

The value of a = 2 > 0 (positive), so the graph of function opens upward and has a minimum value.

The minimum value (y - coordinate of the vertex) is - 25/8.

Step 3 :

Find the y -  intercept :

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

y = 2(0)² - 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, - 2).
• The point paired with it on other side of the axis of symmetry is (- 1, - 2) and has the same y - value.
• Since, The axis of symmetry devides the parabola into two equal parts.So, if thereis a point (1, - 2) 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, - 2) and (1/4, - 2) = 3/4 = The distance between (1/4, - 2) and the point paired with it on other side of the axis of symmetry and has the same y - value.
• The distance between (1/4, - 2) and the point paired with it on other side of the axis of symmetry. = 1/4 - 3/4 = (1 - 3)/4 = - 2/4 = - 1/2.
• Therefore, The point paired with it on other side of the axis of symmetry is (- 1/2, - 2).
• Repeate this, and choose an x - value of 2 and substitute.The new point is at (2, 3).and its corresponding point is (- 3/2, 3).
   
• Connect these points and create a smooth curve.

Graph :