Problem 66 - College Algebra

Question

Begin by graphing the standard quadratic function, \(f(x)-x^{2} .\) Then use transformations of this graph to graph the given function. $$ h(x)--2(x+2)^{2}+1 $$

Step-by-Step Solution

Answer

To graph the function \(h(x)=-2(x+2)^{2}+1\), start by drawing the graph for the standard quadratic function \(f(x)=x^{2}\). Then, shift this graph 2 units to the left, reflect it across the x-axis and stretch it vertically by a factor of 2, and finally shift it 1 unit upwards.

Step 1: Start by graphing the standard quadratic function

The graph of the standard quadratic function \(f(x)=x^{2}\) is a parabola that opens upwards with the vertex at the origin (0,0).

Step 2: Apply horizontal shift

The term inside the bracket, '(x+2)', means that we are shifting the graph of \(x^2\) 2 units to the left. So, every point on the graph of \(x^{2}\) is moved 2 units to the left.

Step 3: Apply Vertical stretch and reflection

The coefficient of \((x+2)^{2}\) is -2. The negative sign indicates a reflection around the x-axis, and the value '2' indicates the graph is vertically stretched by a factor of 2. So, every point (x, y) on the graph obtained after step 2 will become (x, -2y).

Step 4: Apply vertical shift

The term '+1' at the end of the function indicates a vertical shift upwards by 1 unit. Thus, each point (x, y) on the graph obtained after step 3 will now become (x, y+1).

Key Concepts

Standard Quadratic Function

The standard quadratic function is expressed as \( f(x) = x^2 \). It features a graph that is a parabola opening upward, with its vertex located at the origin (0, 0). The 'U' shaped curve is symmetrical, and each point on the graph is squared, which causes the graph to become wider as it moves away from the vertex, forming the characteristic parabola shape.

Understanding this basic form is crucial as it sets the foundation for graphing more complex quadratic functions through various transformations.

Transformations of Quadratic Graphs

Transformations change the position or size of the graph of a function without altering its shape. The general form of a transformed quadratic function is \( h(x) = a(x-h)^2 + k \), representing a vertical stretch or shrink by a factor of 'a', a horizontal shift 'h' units to the right, and a vertical shift 'k' units up.

Using transformations, we can graph a quadratic function from its standard form to a more complex one by applying shifts, stretches, and reflections step by step.

Vertex of a Parabola

The vertex of a parabola represents its maximum or minimum point. In the standard form \( f(x) = x^2 \), the vertex is at the origin (0, 0). However, after transformations, the vertex changes position.

For the vertex form \( h(x) = a(x-h)^2 + k \), the vertex is located at the point (h, k). This information helps determine the initial point from which all transformations are applied, providing a starting point for graphing the altered quadratic function.

Horizontal Shifts in Graphs

Horizontal shifts move the graph of a function left or right from its original position. In the function \( h(x) = a(x-h)^2 + k \), 'h' determines the horizontal shift. If 'h' is positive, the graph shifts 'h' units to the right; if 'h' is negative, the graph shifts 'h' units to the left.

For the function \( h(x) = -2(x+2)^2 + 1 \), the '(x+2)' within the squared term implies a shift of 2 units to the left because it's in the form of 'x - (-2)'.

Vertical Stretches in Graphs

Vertical stretches alter the narrowness or wideness of a parabola without changing its direction. This change is represented by the coefficient 'a' in the quadratic function in vertex form. A value of |a| > 1 will stretch the parabola away from the x-axis, making it narrower.

In the function \( h(x) = -2(x+2)^2 + 1 \), the coefficient -2 indicates that the graph is vertically stretched by a factor of 2 compared to the standard parabola, while the negative sign indicates a reflection, which we will discuss in the next section.

Reflection in Graphs

Reflection in graphs occurs when the graph of a function is mirrored over an axis. For quadratic functions, a negative coefficient 'a' in front of the squared term, such as in \( h(x) = -2(x+2)^2 + 1 \), means the parabola is reflected over the x-axis.

All points that were above the x-axis in the original graph will now be located the same distance below the x-axis, and vice versa, effectively flipping the graph upside down.

Vertical Shifts in Graphs

Vertical shifts move the graph of a function up or down in relation to the y-axis. The value 'k' in \( h(x) = a(x-h)^2 + k \) determines the vertical shift.

For a function like \( h(x) = -2(x+2)^2 + 1 \), '+1' indicates that after applying the horizontal shift, reflection, and vertical stretch, the entire graph shifts upwards by 1 unit. Thus, each point is moved vertically, adding one to its y-coordinate, and locating the vertex 1 unit above the point generated by previous transformations.