Problem 22 - Algebra A Combined Function

Question

Sketch the graph of each function. $$ h(x)=(x+2)^{2}+2 $$

Step-by-Step Solution

Answer

Shift the parabola \(x^2\) 2 units left and 2 units up; vertex is \((-2, 2)\).

Step 1: Identify the Base Function

The base function for this problem is the quadratic function \( f(x) = x^2 \). This is a simple parabola that opens upwards and has its vertex at the origin (0,0).

Step 2: Include the Horizontal Shift

The expression \((x + 2)^2\) indicates a horizontal shift. The \(+2\) inside the parentheses means the graph of \(x^2\) will shift 2 units to the left. Thus, the vertex of the parabola moves from (0,0) to (-2,0).

Step 3: Apply the Vertical Shift

The \(+2\) outside the squared term translates the graph vertically upward by 2 units. Therefore, the vertex, which was at \((-2,0)\) after the horizontal shift, moves to \((-2, 2)\).

Step 4: Sketch the Graph

With the vertex at \((-2, 2)\), draw a parabola that opens upwards. The horizontal line symmetry of the parabola will be the line \(x = -2\). The graph should be a smooth curve and should maintain the shape of the standard quadratic function.

Key Concepts

Horizontal Shift

When dealing with quadratic functions, horizontal shift refers to moving the graph of the function left or right on the coordinate plane. This shift is determined by the term inside the parentheses of the function. For example, in the expression \((x + 2)^2\), the "+2" indicates a horizontal shift. Counterintuitively, a "+" inside the parentheses actually shifts the graph to the left. Therefore, in the function provided, we move the base parabola \(f(x) = x^2\) two units to the left, which updates the vertex from \((0,0)\) to \((-2,0)\).

• The rule is: \( (x + c)^2 \) shifts \( c \) units to the left if \( c \) is positive.
• Conversely, \( (x - c)^2 \) shifts \( c \) units to the right if \( c \) is negative.

Recognizing horizontal shifts allows you to accurately place the function on the graph.

Vertical Shift

A vertical shift in a quadratic function moves the graph up or down along the y-axis. This occurs because of a constant added or subtracted from the quadratic expression. In our example, the \(+2\) located outside the squared term \((x + 2)^2\) indicates a vertical shift upwards by 2 units. This moves the entire graph, including the vertex.After we've performed the horizontal shift, the vertex was at \((-2,0)\). Adding the vertical shift means the vertex relocates to \((-2,2)\). Understanding this behavior enables you to predict and draft the graph's new position conveniently:

• A positive constant (e.g., \(+2\)) shifts the graph up.
• A negative constant would move it down.

Remember, vertical shifts do not affect the horizontal orientation of the parabola. They only impact the y-coordinate position.

Parabola

A parabola is the shape that characterizes the graph of a quadratic function. Typically, it's a symmetrical, U-shaped curve that can either open upwards or downwards. The base form of a parabola can be defined by the simple function \(f(x) = x^2\), which opens upwards from the origin point \((0,0)\).For graphing a function such as \(h(x) = (x + 2)^2 + 2\), the tracing of a parabola will be centered around the vertex, which acts like the 'tip' of this graph. In this specific case:

• The vertex is \((-2, 2)\), found after applying horizontal and vertical shifts.
• The parabola opens upward, because the coefficient of \(x^2\) is positive (1 in this case).
• The line of symmetry is the vertical line \(x = -2\), which passes through the vertex.

Drawing a parabola graph requires attention to its symmetry and the direction it opens. Each point on one side of the vertex should mirror the other side, maintaining the beautiful balance of this mathematical curve.