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Chapter 9: Problem 7

Identify the vertex of each parabola. $$ f(x)=(x-1)^{2} $$

Short Answer

Expert verified
The vertex is at (1, 0).

Step by step solution

01

Recognize the Standard Form

The given function is in the form \(f(x) = (x-h)^{2} + k\). This is the standard form of a parabola where the vertex form is easily identified.
02

Identify h and k

Compare \(f(x) = (x-1)^{2}\) with the standard form \(f(x) = (x-h)^{2} + k\). Here, \(h = 1\) and \(k = 0\), since there is no \(+k\) term.
03

State the Coordinates of the Vertex

Using the values identified for \(h\) and \(k\), the vertex of the parabola is at (h, k). Substituting the values, the vertex is at \( (1, 0) \).

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Key Concepts

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

Standard Form of Parabola
Understanding the standard form of a parabola is crucial when working with quadratic functions. The standard form is given by \( f(x) = (x-h)^2 + k \) where:
  • \(h\) is the x-coordinate of the vertex
  • \(k\) is the y-coordinate of the vertex
This form makes it easy to see the transformation from the parent function \(y = x^2\). The values of \(h\) and \(k\) shift the graph horizontally and vertically, respectively. For example, if \((h, k)\) is (3, -2), the parabola shifts right 3 units and down 2 units from the origin. Identifying these values helps in graphing the parabola and understanding its properties.
Identifying h and k
To identify \(h\) and \(k\), you simply need to compare the given quadratic function to its standard form, \( f(x) = (x-h)^2 + k \). Consider the example given: \( f(x) = (x-1)^2 \)
  • Compare this with \( f(x) = (x-h)^2 + k \).
  • Notice that \( (x-1)^2 \) can be written as \( (x-1)^2 + 0 \).
Therefore, you can identify that \( h = 1 \) and \( k = 0 \). This process is done by matching the given function鈥檚 components to the standard form. Once identified, these values directly lead us to the vertex coordinates.
Vertex Coordinates
Finding the vertex of a parabola in standard form is straightforward once you have identified \(h\) and \(k\). The vertex is simply the point \((h, k)\). For the given function \( f(x) = (x-1)^2 \) We have \( h = 1 \) \( k = 0 \) Therefore, the vertex is at the coordinates \( (1, 0) \) This point represents the minimum or maximum of the parabola, depending on its orientation. For instance, in our example, since there's no negative sign in front of \( (x-1)^2 \), the parabola opens upwards, making the vertex the minimum point of the graph. Identifying the vertex gives insights into the symmetry, direction, and key characteristics of the parabola.

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Most popular questions from this chapter

Use the quadratic formula to solve each equation. (All solutions for these equations are non- real complex numbers.) $$ (2 x-1)(8 x-4)=-1 $$

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ p^{2}+\frac{p}{3}=\frac{1}{6} $$

Graph each parabola. Plot at least two points as well as the vertex. Give the vertex, axis, domain, and range . $$ f(x)=-\frac{1}{2}(x+1)^{2}+2 $$

For each quadratic function, tell whether the graph opens up or down and whether the graph is wider, narrower, or the same shape as the graph of $f(x)=x^{2} . $$ f(x)=-\frac{1}{3}(x+6)^{2}+3 $$

Solve each equation. $$ \sqrt{2 x+6}=x-1 $$
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