Problem 69 - 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 that you can make from your graphs. \(f(x)=x^{2}+c\) (a) \(c=0,2,4,6 ; \quad[-5,5]\) by \([-10,10]\) (b) \(c=0,-2,-4,-6 ; \quad[-5,5]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

Step-by-Step Solution

Answer

The value of \(c\) shifts the graph vertically: up if \(c\) is positive, down if \(c\) is negative, without changing the shape of the parabola.

Step 1: Understanding the Function

The given function is \(f(x) = x^2 + c\). This represents a family of quadratic functions where each function is a parabola opening upwards. The term \(c\) shifts the graph vertically up or down.

Step 2: Graphing for Part (a)

For each specified value of \(c\):- \(c=0\): The parabola \(f(x) = x^2\) has its vertex at the origin (0,0).- \(c=2\): The parabola \(f(x) = x^2 + 2\) shifts 2 units up, with its vertex at (0,2).- \(c=4\): The parabola \(f(x) = x^2 + 4\) shifts 4 units up, with its vertex at (0,4).- \(c=6\): The parabola \(f(x) = x^2 + 6\) shifts 6 units up, with its vertex at (0,6).All these graphs fit within the viewing window of \([-5, 5]\) by \([-10, 10]\).

Step 3: Graphing for Part (b)

For each specified value of \(c\):- \(c=0\): The parabola \(f(x) = x^2\) has its vertex at the origin (0,0).- \(c=-2\): The parabola \(f(x) = x^2 - 2\) shifts 2 units down, with its vertex at (0,-2).- \(c=-4\): The parabola \(f(x) = x^2 - 4\) shifts 4 units down, with its vertex at (0,-4).- \(c=-6\): The parabola \(f(x) = x^2 - 6\) shifts 6 units down, with its vertex at (0,-6).These graphs also fit within the viewing window of \([-5, 5]\) by \([-10, 10]\).

Step 4: Analyzing the Effect of c

The parameter \(c\) in the function \(f(x) = x^2 + c\) affects the vertical position of the parabola on the graph. Increasing \(c\) shifts the graph upwards, and decreasing \(c\) shifts it downwards. The shape of the parabola remains the same regardless of \(c\). This can be concluded from observing the graphical shift of the vertex along the y-axis.

Key Concepts

Graphing Parabolas

When we talk about quadratic functions like \(f(x) = x^2 + c\), we're dealing with a family of curves known as parabolas. These beautiful shapes are always U-shaped when the coefficient of \(x^2\) is positive. A parabola's most distinctive feature is its vertex, which is the point where the curve changes direction.

Graphing a parabola involves finding key features:

• Finding the vertex, which gives us a starting point.
• Identifying symmetry, which helps plot points evenly on either side of the vertex.
• Choosing some x-values to find corresponding y-values for graphing accuracy.

When you plot a few points using these steps and connect them smoothly, the parabola should appear neatly on your graph, demonstrating how quadratic functions translate into graceful curves.

Vertex Form

The vertex form of a quadratic function is a neat way to express a parabola's equation. It typically looks like \(f(x) = a(x-h)^2 + k\). Here:

• \(h\) and \(k\) are the coordinates of the vertex, \((h, k)\).
• The value \(a\) affects the width and direction of the opening.

For our equation, \(f(x) = x^2 + c\), we can see that it matches the vertex form because it indicates the position of the vertex directly. The term \(c\) acts like \(k\) in vertex form, showing us how high or low the vertex sits on the y-axis.Understanding vertex form is very helpful because it instantly tells you the vertex location without needing extra calculations. Knowing where the vertex is allows for easier graph plotting since that's the primary feature of a parabola.

Vertical Shifts

Vertical shifts are all about how graphs can move up and down on a plane. In the equation \(f(x) = x^2 + c\), the \(c\) is the key player in vertical shifts. It's a simple yet powerful element that influences whether the parabola sits higher or lower on the graph.

• If \(c\) is positive, the whole graph moves up by \(c\) units.
• If \(c\) is negative, the graph shifts down by \(c\) units.

What's great about vertical shifts is that they don't change the shape or direction of the parabola—just its position. By adjusting \(c\), you can explore different instances of the same quadratic function's family, seeing how the curve's position changes but not its form.