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Chapter 9: Problem 11

Identify the vertex of each parabola. $$ f(x)=-(x-5)^{2}+6 $$

Short Answer

Expert verified
The vertex is at (5, 6).

Step by step solution

01

Identify the Form of the Equation

The given equation is in the vertex form of a parabola, which is \( f(x) = a(x-h)^2 + k \). Here, \(a\), \(h\), and \(k\) are constants.
02

Extract the Vertex Coordinates

Compare the given equation \( f(x) = -(x-5)^2 + 6 \) with the vertex form. By comparing, it is clear that \( h = 5 \) and \( k = 6 \). Therefore, the vertex coordinates are \((h, k) = (5, 6)\).

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Key Concepts

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

quadratic functions
Quadratic functions are fundamental in algebra and are typically written in the form \[ f(x) = ax^2 + bx + c \]. These functions feature a parabola when plotted on a graph. Quadratic functions can represent various real-world scenarios, like projectile motion and profit maximization. Understanding these functions is essential because they recur in different fields such as physics, engineering, and economics. In the context of our exercise, it is valuable to see how a quadratic function can be rewritten in different forms, like the vertex form, to easily identify certain properties.
vertex form
Vertex form is an alternative way to write a quadratic function, making it easier to identify the vertex of the parabola. The vertex form of a quadratic function is \[ f(x) = a(x-h)^2 + k \], where \( (h, k) \) represents the vertex coordinates. This form is particularly useful because it directly shows the highest or lowest point (the vertex), making it simpler to graph the function. In the given equation \( f(x) = -(x-5)^2 + 6 \), we can see it is already in vertex form, allowing for straightforward extraction of the vertex as \( (5, 6) \).
parabola
A parabola is a U-shaped curve that can open either upwards or downwards. Parabolas have several key features, including the vertex, axis of symmetry, and direction of opening. The vertex is the maximum or minimum point on the parabola. In our example, the vertex is \( (5, 6) \). The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror images. The coefficient \( a \) in the quadratic equation determines the direction of the parabola: if \( a > 0 \), it opens upwards; if \( a < 0 \), it opens downwards. For \( f(x) = -(x-5)^2 + 6 \), \( a \) is negative, indicating it opens downwards.

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Most popular questions from this chapter

Solve each quadratic equation by factoring or by completing the square. $$ x^{2}+3 x-4=0 $$

Graph each parabola. Plot at least two points as well as the vertex. Give the vertex, axis, domain, and range . $$ f(x)=3(x-2)^{2}+1 $$

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ 26 r-2=3 r^{2} $$

What would be the first step in solving \(2 x^{2}+8 x=9\) by completing the square?

Solve each equation. $$ \sqrt{2 x+1}+\sqrt{x+3}=0 $$
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