Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-26: Sketch, on the same coordinate plane, the graphs of \(f\) for the given values of \(c\). (Make use of symmetry, shifting, stretching, compressing, or reflecting.) $$ f(x)=-x^{2}+c ; \quad c=-4,2,4 $$

Step-by-Step Solution

Answer

Graph each parabola \( f(x) = -x^2 + c \) with vertices at (0, -4), (0, 2), and (0, 4).

Step 1: Identify the Basic Function

The given function is in the form \( f(x) = -x^2 + c \). Start with the basic quadratic function \( f(x) = -x^2 \), which is a downward opening parabola centered at the origin (0,0).

Step 2: Introduce Vertical Shift

The value of \( c \) affects the vertical position of the parabola. For \( c = -4 \), the graph is shifted downward by 4 units, placing the vertex at (0, -4). For \( c = 2 \), the vertex is raised 2 units to (0, 2). Finally, for \( c = 4 \), the vertex is at (0, 4).

Step 3: Sketch the Graphs

Draw the basic parabola for \( f(x) = -x^2 \). Then, apply the vertical shifts: move the parabola down to \( (0, -4) \) for \( c = -4 \), to \( (0, 2) \) for \( c = 2 \), and to \( (0, 4) \) for \( c = 4 \). Each resulting curve is symmetric about the y-axis and maintains the same shape, just vertically altered.

Key Concepts

Parabola

A parabola is a specific kind of U-shaped graph representing a quadratic function. It can open upwards or downwards. In our case, we consider a downward-opening parabola because it is described by the function \( f(x) = -x^2 + c \). This function reflects about its highest point, known as the vertex. Parabolas have important properties such as the axis of symmetry, which passes through the vertex. Additionally, they are symmetric about this axis. Understanding these properties is critical in graphing and transforming them correctly.

Graph Transformations

Graph transformations are essential in altering the appearance of a graph without changing its basic shape. For quadratic functions, we perform transformations like translation, reflection, and dilation. In the function \( f(x) = -x^2 + c \), we are primarily concerned with translation. This basic form starts with a reflection over the x-axis, indicated by the negative sign in front of \( x^2 \), which creates a downward-opening parabola. Other transformations might include stretching or compressing, but in this example, we maintain the same width and focus on vertical translation.

Vertical Shifting

Vertical shifting involves moving the graph of a function up or down on the coordinate plane. This is controlled by the constant \( c \) in the quadratic function \( f(x) = -x^2 + c \). It's a simple yet powerful way to adjust the function's position.

• For \( c = -4 \): the entire parabola moves down 4 units, placing the vertex at (0, -4).
• For \( c = 2 \): the parabola moves up 2 units, making the vertex (0, 2).
• For \( c = 4 \): it shifts further up, with the vertex at (0, 4).

Vertical shifting does not alter the shape of the graph; only its position changes.

Symmetry

Symmetry is a key attribute of parabolas. The function \( f(x) = -x^2 + c \) exhibits symmetry about the y-axis. This means that if you fold the graph along the y-axis, both sides will match perfectly. This property is due to it being an even function, where the equation remains unchanged when \( x \) is replaced with \(-x\). Understanding symmetry helps in predicting the behavior of the graph on either side of the line of symmetry. This makes sketching much simpler, as you only need to focus on one half of the parabola and reflect it over the axis.