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Chapter 9: Problem 10

Identify the vertex of each parabola. $$ f(x)=(x+5)^{2}-8 $$

Short Answer

Expert verified
The vertex of the parabola is \((-5, -8)\).

Step by step solution

01

Recognize the Vertex Form

Identify the given quadratic equation in the vertex form, which is \(f(x) = a(x-h)^2 + k \). The equation \( f(x) = (x+5)^2 -8 \) is already in the vertex form.
02

Identify Parameters

Compare the given equation \(f(x) = (x+5)^2 -8 \) with the vertex form \(f(x) = a(x-h)^2 + k \). Here, the coefficient \(a = 1 \), \(h = -5 \) (since \(h= -5 \)), and \(k = -8 \).
03

Determine the Vertex

The vertex of a parabola in the form \( f(x) = a(x-h)^2 + k \) is given by the point \( (h, k) \). Substituting \(h = -5 \) and \(k = -8 \), the vertex is \((-5, -8) \).

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Key Concepts

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

Quadratic Equation
A quadratic equation is a second-order polynomial equation in a single variable, typically written as:

The general form is: \( ax^2 + bx + c = 0 \), where:
  • a is the coefficient of x^2 (the quadratic term)
  • b is the coefficient of x (the linear term)
  • c is the constant term
The graph of a quadratic equation is a parabola, a U-shaped curve that either opens upwards or downwards depending on the sign of the coefficient a. For our exercise, the quadratic equation given is already in the vertex form, which simplifies identifying the vertex.
Vertex Form
The vertex form of a quadratic equation is useful for easily finding the vertex of a parabola. The vertex form is:\[ f(x) = a(x - h)^2 + k \] Here:
  • The parabola's vertex is the point (h,k)
  • a determines the direction and width of the parabola
In our example, we have the equation: \[ f(x) = (x + 5)^2 - 8 \]Comparing this equation with the vertex form, we see:
  • a = 1
  • h = -5 (we take the opposite sign)
  • k = -8
This means the vertex of the parabola is at the point (-5, -8).
Graphing Parabolas
Graphing a quadratic equation involves plotting the parabolic curve on a coordinate plane. Here are the steps to graph a parabola using its vertex form:
  • Identify the vertex (h, k).
  • Determine the direction of the parabola (upward if a > 0, downward if a < 0).
  • Find and plot additional points on either side of the vertex to give the curve its shape.
For our equation \(f(x) = (x + 5)^2 - 8 \), the vertex is (-5, -8), and since the coefficient a is positive (1), the parabola opens upwards. Graphing this parabola will give a symmetric curve with the lowest point at (-5, -8).
Identifying Vertex
Identifying the vertex of a parabola from its equation is straightforward when using the vertex form. Follow these steps:
  • Ensure the quadratic equation is in vertex form, \[ f(x) = a(x - h)^2 + k \].
  • Extract the values of h and k from the equation.
  • The vertex is the point (h, k).
Using the equation \( f(x) = (x + 5)^2 - 8 \):
  • h is -5 (change the sign)
  • k is -8
Thus, the vertex is (-5, -8), which is the turning point of the parabola and provides crucial information for graphing and understanding the behavior of the quadratic function.

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Most popular questions from this chapter

Use the square root property to solve each equation. See Examples \(2,4,\) and \(5 .\) $$ (2-5 t)^{2}=12 $$

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

Working together, two people can cut a large lawn in 2 hr. One person can do the job alone in 1 hr less time than the other. How long (to the nearest tenth) would it take the faster worker to do the job? (Hint: \(x\) is the time of the faster worker.) (TABLE CANNOT COPY)

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

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} . $$ f(x)=\frac{2}{3} x^{2}-4 $$
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