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Magazine Chapter 0: Problem 20
Graph. (Unless directed otherwise, assume that "Graph" means "Graph by hand.") \(y=5-x^{2}\)
Short Answer
Expert verified
The graph of \(y = 5 - x^2\) is a downward-opening parabola with vertex \((0, 5)\) and symmetry about the y-axis.
Step by step solution
01
Identify the Type of Equation
The given equation is \(y = 5 - x^2\). This is a quadratic equation, and its graph will be a parabola. Because the coefficient of \(x^2\) is negative, the parabola opens downwards.
02
Determine the Vertex
The standard form of a parabola is \(y = ax^2 + bx + c\). In our equation, \(a = -1\), \(b = 0\), and \(c = 5\). The vertex form of a parabola is \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex. For the equation \(y = 5 - x^2\), the vertex is at \((0, 5)\).
03
Find the Axis of Symmetry
The axis of symmetry for a parabola in the form \(y = ax^2 + bx + c\) is \(x = -\frac{b}{2a}\). Here, \(b = 0\) and \(a = -1\), so the axis of symmetry is \(x = 0\). This means the parabola is symmetric about the y-axis.
04
Identify the Y-intercept
The y-intercept occurs when \(x = 0\). By substituting \(x = 0\) into the equation, we get \(y = 5 - 0^2 = 5\). Therefore, the y-intercept is at \((0, 5)\).
05
Calculate Additional Points
To get a more accurate graph, calculate a few more points. For example, substitute \(x = 1\) and \(x = -1\) into the equation: - When \(x = 1\), \(y = 5 - 1^2 = 4\).- When \(x = -1\), \(y = 5 - (-1)^2 = 4\).So, additional points include \((1, 4)\) and \((-1, 4)\).
06
Plot the Points and Sketch the Graph
Plot the vertex \((0, 5)\), the y-intercept \((0, 5)\), and the additional points \((1, 4)\) and \((-1, 4)\) on a coordinate plane. Since the parabola opens downwards, draw a smooth curve through these points, ensuring the curve is symmetric about the y-axis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabola Graphing
Graphing a parabola is a crucial skill in understanding quadratic equations. A parabola is a U-shaped curve that expresses the graphical representation of a quadratic equation. In our specific equation, \( y = 5 - x^2 \), the coefficient of the\( x^2 \) term is negative, indicating that the parabola opens downwards. This means the parabola arches like an upside-down letter "U."
- The general rule for identifying the parabola's direction is the sign of the \( a \) coefficient. If \( a \) is positive, the parabola opens upwards. If \( a \) is negative, it opens downwards.
- Key points such as the vertex, y-intercept, and other calculated points help in sketching the parabola accurately on the graph.
- Plot these points on a coordinate plane, and connect them smoothly with a curve to form the parabola.
Vertex of a Parabola
The vertex is the highest or lowest point on a parabola, depending on whether it opens downwards or upwards. In our equation \( y = 5 - x^2 \), the vertex is calculated in the standard form \( y = ax^2 + bx + c \).
The vertex form, which is easier to identify, is \( y = a(x-h)^2 + k \) where \((h, k)\) is the vertex. Here, \( a = -1 \), \( b = 0 \), and \( c = 5 \), making the vertex \((0, 5)\).
The vertex form, which is easier to identify, is \( y = a(x-h)^2 + k \) where \((h, k)\) is the vertex. Here, \( a = -1 \), \( b = 0 \), and \( c = 5 \), making the vertex \((0, 5)\).
- This point \((0, 5)\) is where the parabola reaches its maximum value, as it opens downwards.
- The vertex acts like a pivot, providing symmetry to the parabola.
- Knowing the vertex helps in determining the shape and orientation of the parabola graph.
Axis of Symmetry
The axis of symmetry is an essential feature of parabolas. It is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. For the equation \( y = 5 - x^2 \), we find the axis of symmetry using the formula \( x = -\frac{b}{2a} \).
Since \( b = 0 \) and \( a = -1 \), this results in \( x = 0 \).
Since \( b = 0 \) and \( a = -1 \), this results in \( x = 0 \).
- The equation \( x = 0 \) indicates that the y-axis serves as our axis of symmetry for this parabola.
- This symmetry means that for every point on one side of the axis, there is a corresponding point on the opposite side at the same distance from the axis.
- Plotting the axis of symmetry on your graph helps ensure the parabola is drawn accurately and smoothly.
Y-intercept
The y-intercept is the point where the parabola intersects the y-axis. In our quadratic equation \( y = 5 - x^2 \), we find the y-intercept by setting \( x = 0 \) and solving for \( y \).
Solving gives us \( y = 5 \), which makes the y-intercept \((0, 5)\).
Solving gives us \( y = 5 \), which makes the y-intercept \((0, 5)\).
- The y-intercept provides a specific point for the graph, helping in the initial plotting of the parabola.
- It's a crucial element in sketching because it gives a direct point through which the parabola passes on the y-axis.
- Knowing the y-intercept helps to understand and predict the shape of the parabola, even before drawing it.
