Question

for the quadratic function f(x)= -x² - 2x, answer parts (a) through (f). comma to separate answers as needed.) b. there is/are no x - intercept(s). (c) use parts (a) and (b) to graph the function. use the graphing tool to graph the function. click to enlarge graph

Explanation:

1. Find the x - intercepts (part b):

• The x - intercepts of a function \(y = f(x)\) are found by setting \(y = 0\). So, for the function \(f(x)=-x^{2}-2x\), we set \(f(x) = 0\).
• \(-x^{2}-2x = 0\).
• Factor out \(-x\): \(-x(x + 2)=0\).
• Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\).
• For \(-x=0\), we get \(x = 0\).
• For \(x + 2=0\), we get \(x=-2\).
• The x - intercepts are \(x = 0\) and \(x=-2\).

1. Graphing the function (part c):

• The general form of a quadratic function is \(y = ax^{2}+bx + c\). For \(y=-x^{2}-2x\), \(a=-1\), \(b=-2\), and \(c = 0\).
• The vertex of a quadratic function \(y = ax^{2}+bx + c\) has its x - coordinate given by \(x=-\frac{b}{2a}\).
• Substitute \(a=-1\) and \(b = - 2\) into the formula: \(x=-\frac{-2}{2\times(-1)}=-\frac{-2}{2}=-1\).
• To find the y - coordinate of the vertex, substitute \(x=-1\) into the function \(y=-x^{2}-2x\). So, \(y=-(-1)^{2}-2\times(-1)=-1 + 2=1\). The vertex is \((-1,1)\).
• We know the y - intercept is found by setting \(x = 0\), and when \(x = 0\), \(y = 0\) (from the function \(y=-x^{2}-2x\)).
• We plot the x - intercepts \((-2,0)\) and \((0,0)\), and the vertex \((-1,1)\) and then draw a parabola opening downwards (since \(a=-1<0\)).

Step1: Find x - intercepts

Set \(f(x)=0\), so \(-x^{2}-2x = 0\), factor to \(-x(x + 2)=0\), then \(x = 0\) or \(x=-2\).

Step2: Find vertex

Use \(x=-\frac{b}{2a}\) with \(a=-1\), \(b=-2\) to get \(x=-1\), substitute \(x=-1\) into \(y=-x^{2}-2x\) to get \(y = 1\).

Step3: Identify y - intercept

Set \(x = 0\) in \(y=-x^{2}-2x\), get \(y = 0\).

Step4: Graph

Plot x - intercepts \((-2,0)\), \((0,0)\), vertex \((-1,1)\) and draw a downward - opening parabola.

Answer:

The x - intercepts are \(x = 0\) and \(x=-2\). To graph the function, plot the x - intercepts \((-2,0)\) and \((0,0)\), the vertex \((-1,1)\) and draw a parabola opening downwards.