f(x)=-2(x-1)^2+5

Question

Graphing a Quadratic Function in Standard Form.

 

Answer 1

f(x) = -2(x-1)^2+5

y = -2(x^2+1-2x)+5

y = -2x^2-2+4x+5

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

Now it is in standard form of parabola y = ax^2+bx+c

Graph

 

Answer 2

The function is f(x) = y = - 2(x - 1)² + 5.

Rewrite the function as f(x) = y = - 2x² + 4x + 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 = 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 function, 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 :