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Chapter 2: Problem 34

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$f(x)=-\frac{1}{3} x^{2}+3 x-6$$

Short Answer

Expert verified
The vertex is at \((4.5, 0.75)\), the axis of symmetry is \(x = 4.5\), and the x-intercepts are \(x = 3\) and \(x = 6\). The sketch of the graph would be a downward-opening parabola with the specified vertex, axis of symmetry, and x-intercepts.

Step by step solution

01

Rewrite the equation into standard form

The equation is already given in standard form for a quadratic function which is \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients. So the given function can be written as \(f(x) = -\frac{1}{3}x^2 + 3x - 6\).
02

Identify the vertex

The vertex of a quadratic function is given by \((-\frac{b}{2a}, f(-\frac{b}{2a}))\). In this case, \(a = -\frac{1}{3}\) and \(b = 3\). So the x-coordinate of the vertex is \(-\frac{b}{2a} = -\frac{3}{2*(-\frac{1}{3})} = 4.5\). Substitute \(4.5\) back into \(f(x)\) to get the y-coordinate: \(f(4.5) = -\frac{1}{3}*(4.5)^2 + 3*(4.5) - 6 = 0.75\). So the vertex is \((4.5, 0.75)\).
03

Determine the axis of symmetry

The axis of symmetry of a quadratic function is a vertical line passing through the vertex. So its equation is \(x = -\frac{b}{2a}\). In this case, the axis of symmetry is \(x = 4.5\).
04

Find the x-intercepts

The x-intercepts of the function are the values of \(x\) for which \(f(x) = 0\). So solve the equation \(-\frac{1}{3}x^2 + 3x - 6 = 0\). This is equivalent to solving the quadratic equation \(x^2 - 9x + 18 = 0\). Solving this by factorization gives \(x = 3\) or \(x = 6\), which are the x-intercepts.
05

Sketch the Graph

From the analysis above, it is known the vertex is at \(4.5, 0.75\), and the curve intersects the x-axis at x=3 and x=6, and the axis of symmetry is the vertical line \(x = 4.5\). Since \(a < 0\), the curve opens downwards. Use these characteristics to draw a sketch of the quadratic curve. The exact accuracy of the graph isn't strictly necessary, as the focus here is on understanding the fundamental attributes of the quadratic function.

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