Question

directions: use the attributes provided to sketch a graph and fill out the missing information.

1. domain:
2. range:
3. vertex: (-4, 0)
4. axis of symmetry:
5. direction of opening:
6. p - value:
7. focus: (-6, 0)
8. directrix:
9. write an equation in parabola form:

Explanation:

Step1: Determine the domain

For a parabola that opens horizontally (since the vertex \((-4,0)\) and focus \((-6,0)\) have the same \(y -\)coordinate), the domain is all real numbers. So, Domain: \((-\infty,\infty)\)

Step2: Determine the range

Since the parabola opens horizontally, the \(y -\)values can be any real number. So, Range: \((-\infty,\infty)\)

Step3: Axis of symmetry

For a parabola of the form \((y - k)^2=4p(x - h)\) (horizontal axis), the axis of symmetry is \(y = k\). Given the vertex \((h,k)=(-4,0)\), Axis of symmetry: \(y = 0\)

Step4: Direction of opening

The focus \((-6,0)\) and vertex \((-4,0)\). Since \(h=-4\) and the \(x -\)coordinate of the focus \((x_f=-6)\) with \(x_f

Step5: Calculate \(p\)

The formula for \(p\) is \(p=x_f - h\). Given \(h=-4\) and \(x_f=-6\), \(p=-6-(-4)=-2\)

Step6: Directrix

The formula for the directrix of a horizontal parabola \((y - k)^2 = 4p(x - h)\) is \(x=h - p\). Substituting \(h=-4\) and \(p=-2\), \(x=-4-(-2)=-2\). Directrix: \(x=-2\)

Step7: Equation of the parabola

The standard form of a horizontal parabola is \((y - k)^2=4p(x - h)\). Substituting \(h=-4,k = 0,p=-2\), we get \((y-0)^2=4(-2)(x + 4)\), which simplifies to \(y^{2}=-8(x + 4)\)

Answer:

1. Domain: \((-\infty,\infty)\)
2. Range: \((-\infty,\infty)\)
3. Axis of symmetry: \(y = 0\)
4. Direction of opening: Left
5. \(p - \)value: \(-2\)
6. Directrix: \(x=-2\)
7. Equation: \(y^{2}=-8(x + 4)\)