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Chapter 2: Problem 25

Using Standard Form to Graph a Parabola In Exercises \(17-34\) , write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$ f(x)=x^{2}+34 x+289 $$

Short Answer

Expert verified
The quadratic function in standard form is \(f(x) = (x+17)^2\). The vertex is (-17, 0), the axis of symmetry is \(x = -17\), and the x-intercept is \(x = -17\).

Step by step solution

01

Rewrite the Function in Standard Form

Standard form of a quadratic function is \(f(x) = a(x-h)^2 + k\), where (h, k) is the vertex of the parabola. The given function can be rewritten by completing the square: \(f(x) = (x^2 + 34x + (34/2)^2) - (34/2)^2 + 289 = (x+17)^2 + 0\). So, \(f(x) = (x+17)^2\).
02

Identify the Vertex

The vertex of the parabola, (h, k), is given by the constants in the standard form of the quadratic function. In \(f(x) = (x+17)^2\), the vertex of the parabola is (-17, 0).
03

Identify the Axis of Symmetry

The axis of symmetry can be found by setting the values inside the bracket equal to zero. For \(f(x) = (x+17)^2\), when x + 17 = 0, x = -17. Therefore, the axis of symmetry is \(x = -17\).
04

Find the x-intercept(s)

The x-intercepts are the values of x that make the quadratic function equal to zero. That is, solve \(f(x) = 0\) for \(x\). In the case of \(f(x) = (x+17)^2\), this gives \(0 = (x+17)^2\), from which we find that the only x-intercept is \(x = -17\).

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Key Concepts

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

Standard Form
The standard form of a quadratic function is a convenient way to express a quadratic equation as it allows us to easily identify key features of the parabola. It is written as \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) represents the vertex of the parabola. This form not only highlights the vertex but also provides insight into the parabola's direction and width.
To convert a quadratic function, such as \( f(x) = x^2 + 34x + 289 \), into standard form, we use a method called "completing the square." By rearranging and grouping terms, we rewrite it as \( f(x) = (x+17)^2 + 0 \). This transformation reveals that the vertex is at \( (-17, 0) \), and it simplifies graphing.
Vertex of a Parabola
The vertex of a parabola is a crucial point, marking either its highest or lowest position on the graph, depending on the parabola's orientation.
In the standard form \( f(x) = a(x-h)^2 + k \), the vertex is directly given by \( (h, k) \). For the function \( f(x) = (x+17)^2 \), the vertex is \( (-17, 0) \).
  • For \( a > 0 \), the parabola opens upward, making the vertex the lowest point.
  • For \( a < 0 \), it opens downward, with the vertex as the highest point.
The vertex not only provides significant information about the shape of the parabola but also assists in identifying the axis of symmetry.
Axis of Symmetry
The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex, ensuring the equation of the line is \( x = h \) in the standard form \( f(x) = a(x-h)^2 + k \).
For the quadratic function \( f(x) = (x+17)^2 \), the axis of symmetry is \( x = -17 \). This axis is vital for graphing, as it confirms that for every point \( (x, y) \) on the parabola, there exists a matched point \( (-x + 2h, y) \). The axis of symmetry aids in reducing graphing errors by providing a clear line to center the parabola around.
x-intercepts
The \( x \)-intercepts are points where the parabola intersects the \( x \)-axis. To find them, solve the equation \( f(x) = 0 \) for \( x \). These intercepts are critical as they show where the function has zero value.
When dealing with \( f(x) = (x+17)^2 \), set the function to zero: \((x+17)^2 = 0\). Solving this gives \( x = -17 \), indicating a single \( x \)-intercept at \( (-17, 0) \).
  • If the parabola touches the \( x \)-axis at a single point, it's a repeated root, or the vertex aligns with the intercept.
  • If there are no real \( x \)-intercepts, the parabola doesn't touch the \( x \)-axis and has complex roots.
Knowing the \( x \)-intercepts is beneficial for plotting and offers insights into the function's real-number solutions.

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