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Chapter 4: Problem 37

\(g(x)=-(x+2)(x+8)\)

Short Answer

Expert verified
The roots of the function are at x=-2 and x=-8. The vertex is at \((-5,-1)\). The direction of the parabola is downwards. The y-intercept is at -16.

Step by step solution

01

Find the Roots

The roots of the function are the x-values for which the function equals zero (g(x) = 0). This happens when either (x+2) equals zero, or when (x+8) equals zero.\n\nSo, the roots of the function are: \n1. \(x+2=0\) -> \(x=-2\) \n2. \(x+8=0\) -> \(x=-8\)
02

Find the Vertex

The vertex of a quadratic equation \(ax^2+bx+c\) is given by the formula \((-b/2a, f(-b/2a))\). In the standard form, our function would look like \(g(x)=-x^2-10x-16\). \n\nTherefore, the vertex is located at \(x=-(-10)/2*(-1) = -5\). Using this x value in the function gives us the y of the vertex, which is \(g(-5)=-1\), so the vertex is \((-5, -1)\).
03

Determine the Direction of the Parabola

The coefficient of \(x^2\) determines the direction of the parabola. If the coefficient is positive, the parabola faces upwards, if it is negative, the parabola faces downwards. In this equation, the coefficient of \(x^2\) is -1, so the parabola faces downwards.
04

Determine the y-intercept

The y-intercept of a function is the y-value at x=0. So, we can find it by substituting 0 in place of x in the function: \(g(0) = -(0+2)(0+8) = -16\). Thus, the y-intercept of the function is -16.

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Key Concepts

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

Finding Roots of a Quadratic Equation
Discovering the roots of a quadratic equation is one of the foundational skills in analyzing parabolas. To find these roots, we're essentially asking for which values of 'x' the function equals zero. In the case of the function g(x) = -(x+2)(x+8), we set the function equal to zero and solve for 'x'. We can do this because at the roots, the value of the function is exactly zero. Considering the given quadratic is factored, the roots can be found simply by setting each factor equal to zero:
  • For the first factor, (x+2)=0, the root is x=-2.
  • And for the second factor, (x+8)=0, the root is x=-8.
These roots are the x-values where the graph of the quadratic function crosses the x-axis. Remember, quadratic functions can have either two real roots, one real root (in the case of a perfect square), or no real roots if the graph does not intersect the x-axis.
Y-intercept of a Function
The y-intercept of a function is where the graph of the function crosses the y-axis. In equations, it's the output value when all the input variables are zero. For the function g(x) = -(x+2)(x+8), we find this intercept by evaluating the function at zero: g(0) = -(0+2)(0+8) = -16.

So the y-intercept of our quadratic function is -16. This means that the point (0, -16) is where the graph intersects the y-axis. Identifying the y-intercept is essential when sketching the graph of the function and understanding its overall placement relative to the coordinate plane.

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Most popular questions from this chapter

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