A day full of math games & activities. Find one near you.

Identify the maximum or minimum value and the domain and range of the graph of the function y = 2(x - 2)² - 4.

Solution:

The given function is

y = 2(x - 2)² - 4

y = 2(x² - 4x + 4) - 4

y = 2x² - 8x + 8 - 4

y = 2x² - 8x + 4

We know the graph of an equation y = ax² + bx + c where a ≠ 0 is a parabola. The parabola opens upwards if a > 0 and opens downwards if a < 0. The vertex of the parabola is the point where the axis and parabola intersect. Its x coordinate x = -b/2a and its y coordinate is found out by substituting x = -b/2a in the parabola equation.

The parabola given in the problem statement has a +ve coefficient of x² and hence it is a parabola which opens upwards. Also for the equation
a = 2, b = -8 and c = 4. Therefor the x-coordinate of the vertex is

x = -b/2a

= -(-8)/[2(2)]

= 2

Substituting the value of x = 2 in equation (1) we get,

Now y = 2(2)² - 8(2) + 4

= -4

So the point of the minimum value of the graph of the function is (2, -4).

The graph below verifies the vertex point.

The graph only enables us to establish the domain and range of the function. It is evident from the graph that the Domain of the function is x → (-∞ to +∞), {x, x ∈ R}

and the range is (-4 to +∞), {y, y ≥ -4}