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Chapter 3: Problem 34

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)=x^{2}+4 x-1$$

Short Answer

Expert verified
The roots/intercepts of the function are \(x = -4.47\) and \(x = 0.47\). The vertex is at \((-2,-1)\) and the axis of symmetry is \(x=-2\). The parabola opens upwards. The domain of the function is all real numbers, and the range is \(y \ge -1\).

Step by step solution

01

Determine the Discriminant

The discriminant is given by \(b^{2}-4ac\), where \(a\), \(b\), and \(c\) are coefficients of the quadratic equation. For \(f(x)=x^{2}+4 x-1\), \(a=1\), \(b=4\), and \(c=-1\). Therefore, the discriminant is \(4^{2}-4*1*(-1)=20\). Since the discriminant is positive, there are two distinct real roots.
02

Determine the Intercepts

Set the function equal to zero and solve for \(x\), \(x^{2}+4 x-1=0\). By applying the quadratic formula \(x=\frac{-b \pm \sqrt{d}}{2a}\), we get the roots as \(x=\frac{-4 \pm \sqrt{20}}{2}\). So, the intercepts are \(x = -4.47\) and \(x = 0.47\).
03

Determine the Vertex

The vertex (\(h, k\)) of a quadratic function is given by \((\-frac{b}{2a}, f(-\frac{b}{2a})\) ). For the equation \(f(x)=x^{2}+4 x-1\), \(h=\frac{-4}{2*1}=-2\) and \(k=f(-2)=(-2)^{2}+4*(-2)-1=-1\). So the vertex of the function is \((-2,-1)\).
04

Determine the Axis of Symmetry and the Direction

The axis of symmetry for a parabola is the vertical line \(x=h\), where \(h\) is the x-coordinate of the vertex. So the axis of symmetry is \(x=-2\). Since \(a=1\) is positive, the parabola opens upwards.
05

Determine the Domain and Range

The domain of any quadratic function is all real numbers. The range of a quadratic function that opens upwards is \(y \ge k\), where \(k\) is the y-coordinate of the vertex. So the range of this function is \(y \ge -1\).

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