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Chapter 0: Problem 9

Sketching a Graph by Point Plotting In Exercises \(7-16,\) sketch the graph of the equation by point plotting. \(y=4-x^{2}\)

Short Answer

Expert verified
Once plotted, the graph forms a downward opening parabola that passes through points (-2,0), (-1,3), (0,4), (1,3), and (2,0).

Step by step solution

01

Understand the Equation Type

The given equation is \(y = 4 - x^{2}\). This is a quadratic equation and its graph will be a parabola. The negative sign attached to the \(x^{2}\) term indicates that the parabola opens downwards.
02

Choose the Points to Plot

Select a few values for \(x\) and calculate the corresponding values for \(y\). It's always a good idea to include \(x = 0\) in the chosen values. For continuity and symmetry, it is also a good idea to pick both positive and negative values. For this exercise, let's choose \(x=-2, -1, 0, 1, 2\). Now calculate \(y\) for each point: for \(x = -2\), \(y = 4 - (-2)^{2} = 0\); for \(x = -1\), \(y = 4 - (-1)^{2} = 3\); for \(x = 0\), \(y = 4 - 0 = 4\); for \(x = 1\), \(y = 4 - (1)^{2} = 3\); and for \(x = 2\), \(y = 4 - (2)^{2} = 0\). So, the points to be plotted are (-2,0), (-1,3), (0,4), (1,3), and (2,0).
03

Plot the Points

Mark these points on the coordinate plane. After marking the points, simply draw a soft line that best fits these points. Since it's a quadratic equation, you should form the shape of a parabola
04

Analyze the Graph

Ensure that the graphed parabola opens downward (a property of quadratic equations with a negative \(x^{2}\) coefficient) and touches the points plotted earlier.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equations
Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \). They represent second-degree polynomials because the highest power of the variable \( x \) is squared. In quadratic equations, \( a \), \( b \), and \( c \) are constants with \( a \) not equal to zero. This distinguishes quadratics from linear equations.
  • The standard form of a quadratic equation is \( y = ax^2 + bx + c \).
  • If \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.
  • The solutions or "roots" of quadratic equations can be found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
Understanding quadratic equations is essential because they appear in various real-life situations such as calculating areas, projectile motion, and even in financial forecasting. They form the basis for understanding parabolas, which are crucial in graph sketching.
Graph Sketching
Graph sketching involves visually representing the relationship between variables by marking points on a coordinate plane. It helps to understand how changes in one variable affect another. The process usually involves:
  • Identifying the type of equation you are working with - in our case, a quadratic equation.
  • Selecting a range of \( x \)-values to calculate corresponding \( y \)-values.
  • Plotting the calculated points on a graph.
  • Drawing a smooth curve through the plotted points to complete the sketch.
Graphing is more than just drawing lines; it provides insight into how variables relate to each other. Sketching the graph correctly involves understanding the equation structure, especially the role of each term in defining the curve's shape and direction. Consistent point plotting ensures a more accurate and useful representation.
Parabola
A parabola is the U-shaped graph formed by a quadratic equation. Specifically, it is the graph of a function set by \( y = ax^2 + bx + c \). Parabolas have distinctive features:
  • Vertex: The highest or lowest point on the graph, depending on whether it opens downward or upward. In the equation \( y = 4 - x^2 \), the vertex is the point (0, 4).
  • Axis of Symmetry: A vertical line that divides the parabola into two mirror-image halves. For \( y = 4 - x^2 \), this is the line \( x = 0 \).
  • Direction: If the coefficient of \( x^2 \) is negative, the parabola opens downward, like in our example.
  • Intercepts: Points where the parabola intersects the axes, such as the x-intercepts at (-2,0) and (2,0) or the y-intercept at (0,4).
Understanding the properties of a parabola is crucial for graphing. It helps predict the shape and direction of the curve, ensuring a more precise graph based on the equation provided.

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Most popular questions from this chapter

Sketching the Graph of a Trigonometric Function In Exercises \(55-66,\) sketch the graph of the function. $$y=2 \cos 2 x$$

Sketching a Graph In Exercises \(83-86,\) sketch a possible graph of the situation. $$\begin{array}{l}{\text { A student commutes } 15 \text { miles to attend college. After driving }} \\ {\text { for a few minutes, she remembers that a term paper that is }} \\ {\text { due has been forgotten. Driving faster than usual, she returns }} \\ {\text { home, picks up the paper, and once again starts toward school. }} \\ {\text { Consider the student's distance from home as a function of }} \\ {\text { time. }}\end{array}$$

Sketching the Graph of a Trigonometric Function In Exercises \(55-66,\) sketch the graph of the function. $$y=\csc 2 \pi x$$

Sketching a Graph of a Function In Exercises \(31-38,\) sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. \(f(x)=x+\sqrt{4-x^{2}}\)

Proof Prove that the figure formed by connecting consecutive midpoints of the sides of any quadrilateral is a parallelogram.
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