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Magazine Chapter 2: Problem 15
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.) $$f(x)=-x^{2}+c, \quad c=-4,2,4$$
Short Answer
Expert verified
Graph three parabolas with vertices at (0,-4), (0,2), and (0,4).
Step by step solution
01
Understand the Parent Function
The parent function for our given equations is \( y = -x^2 \). This is a downward-opening parabola with its vertex at the origin \((0,0)\). Understanding this shape helps us determine how transformations affect the graph.
02
Apply Vertical Shifts
The function \( f(x) = -x^2 + c \) represents a vertical shift of the parent function \( y = -x^2 \). We shift the graph vertically by \( c \) units. If \( c > 0 \), the graph moves up; if \( c < 0 \), the graph moves down.
03
Graph for c = -4
For \( c = -4 \), the equation becomes \( f(x) = -x^2 - 4 \). This shifts the entire parabola 4 units downwards. The vertex of this parabola is at \((0, -4)\).
04
Graph for c = 2
For \( c = 2 \), the equation becomes \( f(x) = -x^2 + 2 \). This shifts the entire parabola 2 units upwards. The vertex of this parabola is at \((0, 2)\).
05
Graph for c = 4
For \( c = 4 \), the equation becomes \( f(x) = -x^2 + 4 \). This shifts the entire parabola 4 units upwards. The vertex of this parabola is at \((0, 4)\).
06
Sketch the Graphs
On the same coordinate plane, sketch the parabolas: \( f(x) = -x^2 - 4 \), \( f(x) = -x^2 + 2 \), and \( f(x) = -x^2 + 4 \). Ensure the symmetry about the y-axis is clear for each graph, as they are all vertically shifted versions of the same parabola.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabola
A parabola is a U-shaped curve that can open either upwards or downwards, depending on the function defined. The simplest form of a parabola is represented by the quadratic function \(y = ax^2 + bx + c\). In this context, we're looking at \(y = -x^2 + c\), which is centered on the y-axis and opens downwards due to the negative sign in front of \(x^2\). This iconic shape is characterized by its symmetry, with the y-axis serving as its axis of symmetry. \(c\) dictates the vertical shift, which we'll explore in a later section. A key feature of any parabola is its vertex, the point where it turns, which we'll discuss next.
Vertex
The vertex of a parabola is a crucial point that defines its highest or lowest position on the graph, depending on the parabola's orientation. For our specific function \(y = -x^2 + c\), the vertex is located at \( (0, c)\). Here, the coefficients \(b\) and \(a\) do not affect the vertex position because \(b\) is absent, and for a standard form like \(y = ax^2 + bx + c\), our function is already aligned with the y-axis.
- For \(c = -4\), the vertex is \( (0, -4)\), making it the lowest point because the parabola opens downwards.
- For \(c = 2\), the vertex moves up to \((0, 2)\).
- For \(c = 4\), it reaches \((0, 4)\), indicating further upward shifting.
Vertical Shift
A vertical shift in a function directly impacts the y-values of a graph. In the equation \(y = -x^2 + c\), \(c\) determines the vertical shift. You can think of this as moving the entire parabola up or down along the y-axis, without altering its shape. This is an essential transformation because it helps illustrate the adjustments to the parabola's vertex.
- When \(c < 0\), such as \(c = -4\), the graph shifts downward by 4 units, placing the vertex at \((0, -4)\).
- When \(c = 2\), the graph shifts upward by 2 units, positioning the vertex at \((0, 2)\).
- When \(c = 4\), it moves 4 units upwards, resulting in a vertex at \((0, 4)\).
Graph Sketching
Graph sketching combines all the concepts discussed, allowing you to portray quadratic functions visually. For \(f(x) = -x^2 + c\), sketching involves acknowledging the parabola's symmetry, the vertex's location from its vertical shift, and the shape. To create accurate sketches:
- Start by marking the vertex. This is where your graph will have its highest point along the y-axis.
- Understand that the parabola is symmetric about the y-axis. This means any point on one side has a mirror image on the other.
- Vertical shifts caused by different \(c\) values need to be considered for each graph in the same coordinate plane.
