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Chapter 6: Problem 73

In Exercises 73-76, graph the quadratic function. \(f(x)=(x-4)^{2}+3\)

Short Answer

Expert verified
The graph of the given quadratic function \(f(x)=(x-4)^{2}+3\) is a parabola opening upwards, with the vertex at the point (4, 3).

Step by step solution

01

Identify the vertex of the parabola

The vertex form of a quadratic function is \(f(x)=a(x-h)^{2}+k\), where (h,k) is the vertex of the parabola. So, by comparing our function with the general form, we can recognize the vertex (h, k) of our parabola as (4, 3).
02

Determine the direction of the parabola

The coefficient 'a' in a quadratic function determines the direction of the parabola. If 'a' is positive, the parabola opens upwards, if 'a' is negative, it opens downwards. In our case, 'a' is equal to 1 (as it's not shown), which is a positive number. So, the parabola for our function opens upward.
03

Plot the vertex and sketch the parabola

We start graphing our function by first plotting the vertex point (4, 3) on the coordinate plane. Knowing the parabola opens upwards, draw a parabola shape upwards starting at the vertex. We will get a U-shaped parabola which is the graph of our quadratic function.

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Key Concepts

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

Graphing Parabolas
To understand the visual representation of quadratic functions, it is crucial to grasp the basics of graphing parabolas. A parabola is a symmetric curve and the typical graph of a quadratic function. When graphing, start by identifying critical features such as the vertex, axis of symmetry, and direction of opening. These features aid in accurately sketching the curve on the coordinate plane.
  • Vertex: This point is the peak or the lowest point of the parabola, acting as a turning point.
  • Axis of symmetry: A vertical line that passes through the vertex, splitting the parabola into two mirror-image halves.
  • Direction: Whether the parabola opens upwards or downwards, influenced by the coefficient of the quadratic term.
Plot these on a graph and sketch the general shape of a U or an inverted U.
Vertex Form
The vertex form of a quadratic function makes graphing easier. It is written as \[f(x) = a(x-h)^2 + k\]. This form provides quick insight into the graph's vertex and helps determine the parabolic movement. Here are some elements:

  • (h, k): The Vertex Coordinates

    These values directly show the parabola's highest or lowest point.

  • a: The Leading Coefficient

    Determines the width and direction: it's positive for upwards opening and negative for downwards.
In our example, comparing \(f(x) = (x-4)^2 + 3\) with this form gives vertex (4, 3), making it straightforward to plot the parabola's center point.
Parabola Direction
Understanding the direction of a parabola is key to predicting its curve. The direction depends on the coefficient 'a' in the quadratic expression \(f(x) = a(x-h)^2 + k\). Here's how it affects the graph:
  • If \'a\' is positive, like in \ \((x-4)^2 + 3\) where \'a\' equals 1, the parabola opens upward.
  • If \'a\' is negative, the parabola opens downward, forming a cap-like shape.
This knowledge helps delineate the parabola's path when sketching it on paper.
U-shaped Graph
The term "U-shaped graph" refers to the plot of a quadratic function when it opens upwards. It's one of the most recognizable characteristics of a parabola. When the coefficient \'a\' in the vertex form is positive, as with our function \((x-4)^2 + 3\), the resulting graph will curve upwards, forming a perfect U-shape.
  • The vertex is the bottom or lowest point in this U-shape when the parabola opens upwards.
  • The curvature away from the vertex follows a symmetric path up and out, demonstrating the influence of the quadratic term.
  • Points along the graph continually increase as you move away from the vertex horizontally, confirming it has no maximum point but has a minimum at the vertex.
This U-shape is essential for identifying key behaviors of quadratic functions.

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Most popular questions from this chapter

In the polar coordinate system, if a graph that has symmetry with respect to the pole were folded on the line \(\theta=3 \pi / 4\), the portion of the graph on one side of the fold would coincide with the portion of the graph on the other side of the fold.

In Exercises 79-82, find the exact value of the trigonometric function given that \(u\) and \(v\) are in Quadrant IV and \(\sin u=-\frac{3}{5}\) and \(\cos v=1 / \sqrt{2}\). $$ \sin (u-v) $$

In Exercises 73-78, solve the trigonometric equation. $$ \sqrt{2} \sec \theta=2 \csc \frac{\pi}{4} $$

In Exercises 53-56, use a graphing utility to graph the polar equation and show that the indicated line is an asymptote of the graph. $$ \begin{array}{ccc} \text { Name of Graph } & \text { Polar Equation } & \text { Asymptote } \\ \text { Conchoid } & r=2-\sec \theta & x=-1 \end{array} $$

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Hyperbola } & (4, \pi / 2),(1, \pi / 2) \end{array} $$
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