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Magazine Chapter 14: Problem 46
Sketch the graph of each function. Decide whether each function is one-to-one. \(h(x)=-(x+1)^{2}-4\)
Short Answer
Expert verified
The parabola opens downward with vertex at \((-1, -4)\) and is not one-to-one.
Step by step solution
01
Identifying the Function Type
The given function is \( h(x)=-(x+1)^{2}-4 \). This equation has the form of a quadratic function, specifically a parabola. It can be recognized as a downward-opening parabola because the leading coefficient (the coefficient of \( x^2 \)) is negative.
02
Finding the Vertex
The standard form of a parabola is \( y = a(x-h)^2 + k \). In our function, \( a = -1 \), \( h = -1 \), and \( k = -4 \). The vertex of the parabola is at the point \( (-1, -4) \). This is the highest point on the graph because the parabola opens downward.
03
Determining the Axis of Symmetry
The axis of symmetry of a parabola in the form \( y = a(x-h)^2 + k \) is the vertical line \( x = h \). For this function, the axis of symmetry is \( x = -1 \). This line divides the parabola into mirror images.
04
Analyzing the Shape and Direction
Given \( a = -1 \), the parabola opens downwards. It is vertically stretched compared to \( y = x^2 \) due to the negative sign, causing it to be an inverted U shape.
05
Plotting Key Points
Besides the vertex \((-1, -4)\), choose a couple of points on either side of the vertex to plot. For instance, calculate \(h(0)\) and \(h(-2)\). \(h(0) = -(0+1)^2 - 4 = -1 - 4 = -5\); \(h(-2) = -((-2)+1)^2 - 4 = -1 - 4 = -5\). These points are \((0, -5)\) and \((-2, -5)\), equidistant from the axis of symmetry.
06
Sketching the Graph
Now, sketch the graph using the vertex \((-1, -4)\) and the points found \((0, -5)\) and \((-2, -5)\). Draw the parabola opening downwards through these points, ensuring it is symmetrical about the line \(x=-1\).
07
Checking for One-to-One Property
A function is one-to-one if every y-value corresponds to exactly one x-value. Since this parabola is symmetric and does not pass the Horizontal Line Test (any horizontal line will intersect the parabola more than once), \(h(x)\) is not a one-to-one function.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Function
A quadratic function is a type of polynomial function where the highest degree of the variable is two. It typically takes the form of \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, with \(a eq 0\).
Quadratic functions graph as parabolas, which are U-shaped curves that can open either upwards or downwards. The direction it opens is determined by the sign of \(a\):
Quadratic functions graph as parabolas, which are U-shaped curves that can open either upwards or downwards. The direction it opens is determined by the sign of \(a\):
- If \(a > 0\), the parabola opens upwards.
- If \(a < 0\), it opens downwards, like in the case of our function \(h(x)=-(x+1)^{2}-4\).
Vertex of a Parabola
The vertex of a parabola is a critical point that represents the maximum or minimum value of the function. For parabolas that open downward, like \(h(x)=-(x+1)^{2}-4\), the vertex is the highest point.
The vertex is expressed as \((h, k)\) in the vertex form of a quadratic equation, \(y = a(x-h)^2 + k\). In our example, the vertex is located at \((-1, -4)\). This is determined by:
The vertex is expressed as \((h, k)\) in the vertex form of a quadratic equation, \(y = a(x-h)^2 + k\). In our example, the vertex is located at \((-1, -4)\). This is determined by:
- Rewriting the quadratic in vertex form to identify \(h\) and \(k\).
- Using \((-1, -4)\) to indicate the location where the parabola's direction changes.
Axis of Symmetry
The axis of symmetry is an essential feature of a quadratic function's graph. It is a vertical line that runs through the vertex, dividing the parabola into two congruent halves.
For the standard vertex form \(y = a(x-h)^2 + k\), the axis of symmetry is given by the equation \(x = h\).
For the standard vertex form \(y = a(x-h)^2 + k\), the axis of symmetry is given by the equation \(x = h\).
- In our specific function \(h(x)=-(x+1)^{2}-4\), the axis of symmetry is \(x = -1\).
- This line ensures that each point on one side of the parabola has a corresponding point on the opposite side, making the function symmetric about this line.
One-to-One Function
A one-to-one function is a type of function where each \(y\)-value corresponds to exactly one \(x\)-value. This means the function passes the Horizontal Line Test鈥攁ny horizontal line should intersect the graph at most once.
Unfortunately, quadratic functions like \(h(x)=-(x+1)^{2}-4\) are not one-to-one due to their symmetrical, U-shaped graphs.
Unfortunately, quadratic functions like \(h(x)=-(x+1)^{2}-4\) are not one-to-one due to their symmetrical, U-shaped graphs.
- The horizontal line can easily pass through more than one point on a parabola.
- Because of this symmetry, parabolas have two \(x\)-values for most \(y\)-values, failing the one-to-one criteria.
