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Chapter 2: Problem 31

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$h(x)=4 x^{2}-4 x+21$$

Short Answer

Expert verified
The standard form of the provided quadratic function is \(h(x) = 4(x - 1/2)^2 + 19\). The vertex of this function is \((1/2, 19)\), the axis of symmetry is \(x=1/2\), and it has no \(x\)-intercepts.

Step by step solution

01

Rewrite the function in standard form

To transform the given quadratic function into standard form, we will use the method of completing the square. \n \(h(x) = 4x^2 - 4x + 21\)\n Firstly, factor out the coefficient \(4\) from the first two terms: \(h(x) = 4(x^2 - x) + 21\)\n Next, rewrite \(-x\) as \(-2*(x - 1/2)\) and complete the square inside the brackets: \(h(x) = 4[(x - 1/2)^2 - 1/4] + 21\)\n Simplify the equation to find the standard form of the function: \(h(x) = 4(x - 1/2)^2 + 19\)
02

Identify the vertex and axis of symmetry

The vertex form of the equation is \(h(x)=a(x-h)^2+k\), where \((h, k)\) is the vertex of the quadratic function. From the obtained equation \(h(x) = 4(x - 1/2)^2 + 19\), we can identify that the vertex of the function is \((1/2, 19)\). The axis of symmetry is the vertical line \(x=h\), so for this function, the axis of symmetry is \(x=1/2\).
03

Find the x-intercept(s)

The x-intercepts of the function are found by setting \(h(x)\) equal to zero and solving for \(x\). So, we will solve the equation \(4(x - 1/2)^2 + 19 = 0\). However, since the term inside the square is squared and added to the positive constant 19, this equation will not have any real solutions. That means this function has no x-intercepts.

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