Problem 27 - Algebra for College Students

Question

Graph each quadratic function, and state its domain and range. $$h(x)=(x-2)^{2}$$

Step-by-Step Solution

Answer

Domain: (-∞, ∞). Range: [0, ∞).

Step 1 - Identify the quadratic function form

First, note that the given function is in the form of h(x) = (x-2)^2, which is a standard quadratic function written as y = (x-h)^2.

Step 2 - Determine the vertex

Identify the vertex of the quadratic function. In the function h(x)=(x-2)^2, the vertex is at the point (2, 0). This is because the function takes the form (x-h)^2 + k, where the vertex is (h, k).

Step 3 - Plot the vertex on the graph

Plot the vertex (2,0) on the coordinate plane.

Step 4 - Sketch the parabola

Since the coefficient of the squared term is positive, the parabola opens upwards. Sketch the graph of the parabola ensuring it passes through the vertex and opens upwards.

Step 5 - Determine the domain of the function

The domain of any quadratic function is all real numbers because there are no restrictions on the values of x. Therefore, the domain is (-∞, ∞).

Step 6 - Determine the range of the function

Since the parabola opens upwards and the vertex is the lowest point, the range consists of all y values greater than or equal to the y-coordinate of the vertex (which is 0). Therefore, the range is [0, ∞).

Key Concepts

vertex

The vertex of a quadratic function is a very important point. It represents the maximum or minimum point of the parabola. In the given function, = (x - 2)^2), the vertex is at (2, 0). This point is found by identifying the values of in the vertex form <(x - h)^2 + k>. Here, is 2 and is 0.

• For the function = (x - 2)^2, we see that it has its minimum value at the vertex.
• This vertex is crucial for sketching the graph because it gives us a reference point.
• From the vertex, we know the direction the parabola will open (upwards in this case).

The vertex helps us understand the shape and position of the parabola in the coordinate plane.

domain

The domain of a quadratic function is the set of all possible input values (x-values). It tells us which x-values we can plug into the function to get a valid y-value. For any quadratic function, the domain is always all real numbers. This is because there are no x-values that would make the function undefined.

• In our specific example <(h(x) = (x - 2)^2>, we can input any x value into the function and calculate a corresponding y-value.
• This is why we say the domain is <(-∞, ∞/>

So, whatever x-value you choose, it always yields a valid result for the quadratic function.

range

The range of a quadratic function is the set of all possible output values (y-values). For the given function, <(h(x) = (x - 2)^2>, the range is determined based on the vertex and the direction of the parabola. Because our vertex is (2, 0) and the parabola opens upwards (as indicated by the positive coefficient of the squared term), the y-values start from the y-coordinate of the vertex and go upwards.

• This means that the smallest y-value is 0 (the vertex point).
• All other y-values are greater than 0

Therefore, the range of the function is <[0, ∞]).

parabola

A parabola is the U-shaped curve that represents a quadratic function graphically. The parabola can open upwards or downwards. In the given function, <(h(x) = (x - 2)^2>, the parabola opens upwards since the coefficient of the squared term is positive.

• The vertex (2, 0) is the lowest point on this parabola.
• Every point on the parabola is equidistant from a fixed point (the focus) and a fixed line (the directrix), but for easier understanding, it's enough to know the basics of the shape and direction.

Knowing the properties of parabolas helps us sketch the graph accurately and understand the behavior of the function.
Specifically, it helps us determine the direction of the opening and the relative position of all points on the curve.