Problem 74 - Algebra and Trigonometry

Question

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions you can make from your graphs. $$f(x)=(x-c)^{2}$$ (a) \(c=0,1,2,3 ;[-5,5]\) by \([-10,10]\) (b) \(c=0,-1,-2,-3 ; \quad[-5,5]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

Step-by-Step Solution

Answer

The parameter \( c \) shifts the parabola horizontally along the x-axis.

Step 1: Understand the function

The given family of functions is of the form \( f(x) = (x-c)^2 \), where \( c \) is a parameter that shifts the graph horizontally along the x-axis. The graph is a parabola that opens upwards with its vertex at \( (c, 0) \).

Step 2: Graph for (a) c = 0, 1, 2, 3

For each value of \( c \), plot the function in the viewing window \([-5,5]\) by \([-10,10]\).- **For \( c = 0 \)**: The function is \( f(x) = x^2 \), with its vertex at (0, 0).- **For \( c = 1 \)**: The function becomes \( f(x) = (x-1)^2 \), with its vertex at (1, 0).- **For \( c = 2 \)**: The function is \( f(x) = (x-2)^2 \), with its vertex at (2, 0).- **For \( c = 3 \)**: The function becomes \( f(x) = (x-3)^2 \), with its vertex at (3, 0).All graphs are parabolas opening upwards.

Step 3: Graph for (b) c = 0, -1, -2, -3

Again, plot each function in the specified viewing window.- **For \( c = 0 \)**: The function is \( f(x) = x^2 \), with its vertex at (0, 0).- **For \( c = -1 \)**: The function is \( f(x) = (x+1)^2 \), with its vertex at (-1, 0).- **For \( c = -2 \)**: The function becomes \( f(x) = (x+2)^2 \), with its vertex at (-2, 0).- **For \( c = -3 \)**: The function is \( f(x) = (x+3)^2 \), with its vertex at (-3, 0).All these graphs are also parabolas opening upwards.

Step 4: Draw conclusions from the graphs

From the graphs for parts (a) and (b), observe that the parameter \( c \) in the function \( f(x) = (x-c)^2 \) controls the horizontal shift of the parabola along the x-axis. A positive \( c \) shifts the parabola to the right, while a negative \( c \) shifts it to the left. The shape and orientation of the parabolas remain the same.

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions characterized by an equation of the form \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). These equations graph as parabolas, which are U-shaped curves. Parabolas have a special point called the vertex, which is the highest or lowest point of the curve, depending on the orientation. For upward-opening parabolas, which occur when \( a > 0 \), the vertex is the lowest point. Conversely, for downward-opening parabolas when \( a < 0 \), the vertex is the highest point.

• The axis of symmetry is a vertical line passing through the vertex, dividing the parabola into two mirror-image halves.
• The general form of \( f(x) = ax^2 + bx + c \) can be converted to vertex form, \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex.
• Quadratic functions exhibit symmetry and have important practical applications, such as in physics and engineering.

Horizontal Shifts

A horizontal shift is a transformation applied to the graph of a function, moving it left or right on the Cartesian plane. For quadratic functions in the form \( f(x) = (x-c)^2 \), the value of \( c \) dictates this shift.

• A positive \( c \) value results in the graph moving \( c \) units to the right.
• A negative \( c \) value shifts the graph \( |c| \) units to the left.
• The shape of the parabola remains unchanged during a horizontal shift; only its position alters.

This means that if you start with the basic quadratic function \( x^2 \) and alter it to \( (x-c)^2 \) or \( (x+c)^2 \), the graph simply slides along the x-axis but keeps its width, orientation, and vertex form the same. Understanding horizontal shifts is crucial for predicting changes in position without recalculating individual points.

Vertex Form

The vertex form of a quadratic function is expressed as \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. This form is particularly useful for easily identifying the vertex, which simplifies sketching the graph.

• The "\( a \)" in the vertex form influences the width and direction of the parabola. A larger absolute value of \( a \) results in a narrower parabola, while a smaller value results in a wider one.
• The vertex \((h, k)\) is the point where the parabola changes direction.
• Vertex form also makes it easier to implement horizontal and vertical shifts without modifying the function's general structure.

Converting quadratic expressions into vertex form involves completing the square, an algebraic technique that rearranges the expression to highlight this vertex structure. This form is beneficial for transformations and in situations requiring quick visual insights into a parabola's traits.