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Chapter 2: Problem 20

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$g(x)=x^{2}+2 x+1$$

Short Answer

Expert verified
The quadratic function in standard form is \(g(x) = (x + 1)^2\). The vertex is at (-1, 0), the axis of symmetry is at \(x = -1\), and there is one \(x\)-intercept at (-1, 0).

Step by step solution

01

Rewrite the function in standard form

To obtain the standard form, the given function will be written in the form \(a (x - h)^2 + k\). This can be done by completing the square: \(g(x) = x^2 + 2x + 1 = (x + 1)^2 + 0\).
02

Identify the vertex

The vertex of the quadratic function in standard form is given by the point \((h, k)\). Here, the vertex is \((-1, 0)\).
03

Identify the axis of symmetry

The axis of symmetry is a vertical line that passes through the vertex. Its equation is \(x = h\). Thus, the axis of symmetry here is \(x = -1\).
04

Identify the x-intercepts

The \(x\)-intercept(s) can be found by setting \(g(x) = 0\), and then solving the resulting equation. If \(g(x) = 0\) and \(g(x) = (x + 1)^2 + 0\), the \(x\)-intercept is given by the solution of \((x + 1)^2 = 0\), which results in \(x = -1\). Consequently, there is just one \(x\)-intercept at \((-1, 0)\).

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Most popular questions from this chapter

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