Question

graph the equation. y = -2(x - 1)^2 - 4

Explanation:

Step1: Identify the parent function and transformations

The given equation is \( y = -2(x - 1)^2 - 4 \). The parent function is \( y = x^2 \), which is a parabola opening upwards with vertex at \((0,0)\).

• The \((x - 1)\) inside the square indicates a horizontal shift. The formula for horizontal shift is: if we have \( y = (x - h)^2 \), the vertex shifts to \((h,0)\). Here, \( h = 1 \), so the vertex shifts 1 unit to the right.
• The coefficient \(-2\) in front of the square term: the negative sign reflects the parabola over the \( x \)-axis (so it opens downwards), and the absolute value \( 2 \) vertically stretches the parabola by a factor of \( 2 \).
• The \(-4\) at the end is a vertical shift. For \( y = (x - h)^2 + k \), the vertex is \((h,k)\). Here, \( k = -4 \), so the vertex shifts 4 units down.

Step2: Determine the vertex

From the vertex form \( y = a(x - h)^2 + k \), the vertex is \((h,k)\). For \( y = -2(x - 1)^2 - 4 \), \( h = 1 \) and \( k = -4 \), so the vertex is \((1, -4)\).

Step3: Find the direction and width

Since \( a = -2 \), which is negative, the parabola opens downwards. The absolute value of \( a \) is \( 2 \), which is greater than \( 1 \), so the parabola is narrower than the parent function \( y = x^2 \).

Step4: Plot the vertex and use the stretch factor to find other points

• Start with the vertex \((1, -4)\).
• To find other points, we can use the stretch factor. For the parent function \( y = x^2 \), when \( x = 0 \), \( y = 0 \); when \( x = 2 \), \( y = 4 \). But with the transformations:
• For \( x = 0 \):
• Substitute \( x = 0 \) into the equation: \( y = -2(0 - 1)^2 - 4 = -2(1) - 4 = -2 - 4 = -6 \). So the point is \((0, -6)\).
• For \( x = 2 \):
• Substitute \( x = 2 \) into the equation: \( y = -2(2 - 1)^2 - 4 = -2(1) - 4 = -2 - 4 = -6 \). So the point is \((2, -6)\).
• For \( x = -1 \):
• Substitute \( x = -1 \) into the equation: \( y = -2(-1 - 1)^2 - 4 = -2(4) - 4 = -8 - 4 = -12 \). So the point is \((-1, -12)\).
• For \( x = 3 \):
• Substitute \( x = 3 \) into the equation: \( y = -2(3 - 1)^2 - 4 = -2(4) - 4 = -8 - 4 = -12 \). So the point is \((3, -12)\).

Step5: Sketch the parabola

Plot the vertex \((1, -4)\) and the other points we found \((0, -6)\), \((2, -6)\), \((-1, -12)\), \((3, -12)\). Then draw a smooth curve through these points, making sure it opens downwards and is narrower than the parent function \( y = x^2 \).

Answer:

To graph \( y = -2(x - 1)^2 - 4 \):

1. Vertex: \((1, -4)\) (plot this point).
2. Direction: Opens downward (due to the negative coefficient \(-2\)).
3. Stretch/Compression: Vertically stretched by a factor of \( 2 \) and reflected over the \( x \)-axis.
4. Additional Points: Use the vertex and the stretch factor to find symmetric points (e.g., \((0, -6)\), \((2, -6)\), \((-1, -12)\), \((3, -12)\)) and draw a smooth parabola through them.

(Note: Since the problem asks to graph the equation, the final answer is the graphical representation as described above, with the vertex at \((1, -4)\), opening downward, stretched by 2, and passing through the additional points calculated.)