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Chapter 9: Problem 8

Identify the vertex of each parabola. $$ f(x)=(x+3)^{2} $$

Short Answer

Expert verified
The vertex is \((-3, 0)\).

Step by step solution

01

Identify the Standard Form

The standard form for a parabola written is given by the equation \(f(x) = a(x-h)^2 + k\), where \( (h, k) \) is the vertex of the parabola.
02

Make the Equation Match the Standard Form

Compare the given equation \(f(x) = (x + 3)^2\) to the standard form \(f(x) = a(x - h)^2 + k\). Notice that it can be rewritten as \(f(x) = (x - (-3))^2 + 0\).
03

Identify the Vertex

From the rewritten equation \(f(x) = (x - (-3))^2 + 0\), it is clear that \(h = -3\) and \(k = 0\). Therefore, the vertex of the parabola is \((-3, 0)\).

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Key Concepts

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

standard form of a parabola
Understanding the standard form of a parabola is key to solving many quadratic equations. The standard form is given by the equation: f(x) = a(x-h)^2 + k In this equation,
  • a
    • is a coefficient that affects the width and direction of the parabola.
  • (h, k)
    • represents the coordinates of the vertex.
  • If a is positive, the parabola opens upwards. If a is negative, it opens downwards.
To easily identify the vertex, the goal is to write any given quadratic in this standard form.
identifying the vertex
The vertex of a parabola is the highest or lowest point on its graph. This point gives crucial information about the graph's shape and location. To find the vertex from the standard form equation f(x) = a(x-h)^2 + k :
  • Identify the values of h and k; these are the vertex's coordinates.
  • For the equation f(x) = (x + 3)^2, rewrite it in the standard form: f(x) = (x - (-3))^2 + 0
From this, we see that - h = -3 - k = 0 So, the vertex is (-3, 0).
transforming equations
Transforming equations into the standard form is often a necessary step in identifying key features of the parabola, such as its vertex. Here's how to transform an equation:
  • Start with the given quadratic equation, for example, f(x) = (x + 3)^2.
  • Compare it to the standard form, f(x) = a(x-h)^2 + k, and rewrite accordingly: f(x) = (x - (-3))^2 + 0.
This makes it easy to see the vertex. Always aim to rewrite each term in a way that matches the standard form. This involves completing the square when necessary, and identifying the values of h and k.

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Most popular questions from this chapter

Find the vertex of each parabola. For each equation, decide whether the graph opens up. down, to the left, or to the right, and whether it is wider, narrower, or the same shape as the graph of \(y=x^{2} .\) If it is a parabola with vertical axis, find the discriminant and use it to determine the number of \(x\) -intercepts. See Examples \(1-3,5,8,\) and 9 $$ f(x)=2 x^{2}+4 x+5 $$

Graph each parabola. Plot at least two points as well as the vertex. Give the vertex, axis, domain, and range . $$ f(x)=\frac{4}{3}(x-3)^{2}-2 $$

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

A washing machine can be filled in 6 min if both the hot and cold water taps are fully opened. Filling the washer with hot water alone takes 9 min longer than filling it with cold water alone. How long docs it take to fill the washer with cold water?

The function given in Exercise 47 becomes \(D(t)=13 t^{2}-73 t\) for a car going \(50 \mathrm{mph}\) Find the time it takes for this car to skid 218 ft.
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