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Chapter 2: Problem 3

\(f(x)=(x-3)^2\)

Short Answer

Expert verified
The function \(f(x) = (x - 3)^2\) is a quadratic function in the form \(f(x) = a(x - h)^2 + k\), the vertex of the parabola is at the point (3, 0), and since the coefficient of the square term is positive, the function increases on either side of the vertex.

Step by step solution

01

Identify the Form

The function \(f(x) = (x - 3)^2\) follows the form of quadratic function which is \(f(x) = a(x - h)^2 + k\). This form of quadratic function makes it easy to identify the vertex of the parabola, where \(h\) and \(k\) are the x and y coordinates of the vertex, respectively. In this case, \(h = 3\) and \(k = 0\).
02

Identify the Vertex

Using the form of the function, we can see that the vertex of the parabola is at the point (3, 0). This means the parabola opens upwards and has its lowest point (in this case the vertex) at (3,0).
03

Identify the Direction

Since the coefficient of the square term is positive, this parabola opens upwards. This means that the function \(f(x)\) increases on either side of the vertex (3,0).

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Key Concepts

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

Vertex Form
The vertex form of a quadratic function is an essential concept because it helps us easily determine key characteristics of the graph, like the vertex. A quadratic function in vertex form looks like this:
  • \( f(x) = a(x-h)^2 + k \)
  • Here, \(a\), \(h\), and \(k\) are constants.
The vertex
  • is a significant point on the parabola represented by the coordinates \((h, k)\).
  • The value "\(h\)" shifts the parabola horizontally, and "\(k\)" shifts it vertically.
  • If \(a > 0\), the parabola opens upward, and if \(a < 0\), it opens downward.
In our exercise, with \(f(x) = (x-3)^2\), the vertex form allows us to identify the vertex at \((3,0)\).
This gives us valuable information about the function's lowest point.
Parabola
A parabola is a U-shaped curve that is the graphical representation of a quadratic function. This shape has several important features:
  • The vertex, which is the highest or lowest point, depending on direction.
  • The axis of symmetry, a vertical line that passes through the vertex.
Understanding a parabola can help you analyze many real-world situations, such as projectile motion or optimizing areas.
The vertex serves as a turning point of the parabola鈥攙alues to the left and right of the vertex determine the general shape of the parabola.
The exercise shows our parabola opens upward with a vertex at (3,0), because the term \((x-3)^2\) has a positive coefficient.
Graphing Quadratic Functions
Graphing quadratic functions using their vertex form can be straightforward once you know the key components, like the vertex and direction of the parabola. Here鈥檚 a step-by-step approach:
  • Start by identifying the vertex. For \(f(x) = (x-3)^2\), the vertex is \((3,0)\).
  • Identify the direction. Since the coefficient of the \((x-h)^2\) term is positive, the parabola opens upwards.
  • Consider additional points: Choose a few x-values around the vertex and compute their corresponding y-values to plot more points on the graph.
  • Draw the axis of symmetry. This is a vertical line going through \(x = h\), so for our function, it鈥檚 \(x = 3\).
  • Complete the graph by drawing a smooth curve through the points you've plotted, making sure the curve is symmetrical around the axis of symmetry.
This simple guide can help you visualize the quadratic function effectively, allowing for a better understanding of its behavior. Utilizing these steps makes the process of graphing more intuitive and accurate.

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Most popular questions from this chapter

\(h(x)=-5(x+5)(x+1)\)

\(\sqrt{5 x}+5=0\)

The path of a basketball thrown at an angle of \(45^{\circ} \mathrm{can}\) be modeled by \(y=-0.02 x^2+x+6\).

reflection in the \(y\)-axis followed by a translation 4 units right $$ \begin{aligned} h(x) &=f(-x) \\ &=2(-x)^2+6(-x) \\ &=2(x)^2-6 x \\ g(x) &=h(x-4) \\ &=2(x-4)^2+6(x-4) \\ &=2 x^2-10 x+8 \end{aligned} $$

USING STRUCTURE Which function represe the parabola with the widest graph? Explain your reasoning. (A) \(y=2(x+3)^2\) (B) \(y=x^2-5\) (C) \(y=0.5(x-1)^2+1\) (D) \(y=-x^2+6\)
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