Questions: Graph the following function. y=-2(x+3)^2+5

Transcript text: Graph the following function. \[ y=-2(x+3)^{2}+5 \]

Solution

Solution Steps

Step 1: Identify the function and its components

The given function is \( y = -2(x + 3)^2 + 5 \). This is a quadratic function in vertex form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.

Step 2: Determine the vertex

In the function \( y = -2(x + 3)^2 + 5 \), the vertex form is \( y = a(x - h)^2 + k \). Here, \( h = -3 \) and \( k = 5 \). Therefore, the vertex of the parabola is \( (-3, 5) \).

Step 3: Identify the direction and shape of the parabola

The coefficient \( a = -2 \) indicates that the parabola opens downwards (since \( a \) is negative) and is vertically stretched by a factor of 2.

Step 4: Plot the vertex and additional points

Plot the vertex at \( (-3, 5) \).
Choose additional points to plot by substituting values of \( x \) into the function. For example:
- For \( x = -2 \): \( y = -2(-2 + 3)^2 + 5 = -2(1)^2 + 5 = -2 + 5 = 3 \)
- For \( x = -4 \): \( y = -2(-4 + 3)^2 + 5 = -2(-1)^2 + 5 = -2 + 5 = 3 \)

Step 5: Draw the parabola

Using the vertex and the additional points, draw the parabola opening downwards.

Final Answer

The graph of the function \( y = -2(x + 3)^2 + 5 \) is a downward-opening parabola with a vertex at \( (-3, 5) \) and passing through points such as \( (-2, 3) \) and \( (-4, 3) \).

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