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Chapter 11: Problem 8

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

Short Answer

Expert verified
The vertex is (0,0).

Step by step solution

01

Identify the General Form

The general form of a quadratic function is given by \[f(x) = ax^2 + bx + c\] where \( a \), \( b \), and \( c \) are constants.
02

Compare with the Given Function

Compare the given function \( f(x) = \frac{1}{2} x^2 \) with the general form. Here, \( a = \frac{1}{2} \), \( b = 0 \), and \( c = 0 \).
03

Calculate the Vertex Coordinates

The vertex form of a quadratic function is \[ f(x) = a(x-h)^2 + k \] where \( (h, k) \) is the vertex. For the vertex calculation in standard form, the formula for the vertex \( h \) (or x-coordinate) is \( h = -\frac{b}{2a} \). Given \( b = 0 \) and \( a = \frac{1}{2} \), we find: \[ h = -\frac{0}{2 \times \frac{1}{2}} = 0 \]. Then, substitute \( 0 \) for \( x \) in the function to find \( k \): \[ k = f(0) = \frac{1}{2} (0)^2 = 0 \]. Thus, the vertex is \( (0, 0) \).

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Key Concepts

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

Quadratic Functions
Quadratic functions describe the relationship between the variables in the form of a parabola. Every quadratic function can be written in the form \( f(x) = ax^2 + bx + c \). Here:

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Most popular questions from this chapter

Solve using the square root property. Simplify all radicals. $$ (5 t+3)^{2}=36 $$

One solution of \(4 x^{2}+b x-3=0\) is \(-\frac{5}{2} .\) Find \(b\) and the other solution.

One solution of \(3 x^{2}-7 x+c=0\) is \(\frac{1}{3} .\) Find \(c\) and the other solution.

Use the quadratic formula to solve each equation. (All solutions for these equations are non real complex numbers.) $$ z(2 z+3)=-2 $$

Find the discriminant. Use it to determine whether the solutions for each equation are A. two rational numbers B. one rational number C. two irrational numbers D. two nonreal complex numbers. Tell whether the equation can be solved using the zero-factor property, or if the quadratic formula should be used instead. Do not actually solve. $$ 9 x^{2}-12 x-1=0 $$
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