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The vertex form of a parabola is a way to easily identify the vertex, which is a crucial feature of the graph. The formula for the vertex form is given by \( f(x)=a(x-h)^2+k \), where \((h, k)\) is the vertex of the parabola. Here:

• \( a \) determines the direction and the width of the parabola.
• \( h \) is the x-coordinate of the vertex.
• \( k \) is the y-coordinate of the vertex.

So, in the equation \( f(x)=3(x+2)^2-5 \):

• \( a = 3 \)
• \( h = -2 \)
• \( k = -5 \)

This tells us that the vertex of the parabola is at \((-2, -5)\), indicating that the graph opens upwards because \( a > 0 \). Understanding this form helps in quickly sketching or mentally visualizing the parabola by plotting the vertex and considering the stretching determined by \( a \).

Symmetry is a defining feature of parabolas. A parabola is always symmetric about a vertical line called the axis of symmetry. For a parabola in vertex form \( f(x) = a(x-h)^2 + k \), the axis of symmetry is the vertical line \( x = h \). This means that for every point on one side of the parabola, there is a corresponding point on the other side that is equidistant from the axis of symmetry.
In our exercise, the axis of symmetry is given by \( x = -2 \). This allows you to predict or find other points on the parabola. If you know one point, such as \((-1, -2)\), you can find its "mirror" point by measuring its distance from the axis. Since \(-1\) is one unit away from \(-2\), you can move an equivalent distance in the opposite direction to get to \(-3\). Thus, the point \((-3, -2)\) mirrors \((-1, -2)\) across the line \( x = -2 \). This property is especially handy in graphing the parabola and ensuring its shape is accurate.

When graphing a parabola, understanding its different parameters simplifies the process greatly. Each element of the vertex form equation \( f(x)=a(x-h)^2+k \) plays a role in shaping the parabola:

• The vertex \((h, k)\) is the starting point of your graph.
• The axis of symmetry \(x = h\) divides the parabola into two mirror-image halves.
• The value of \( a \) affects the width and orientation of the parabola.

To sketch the graph of \( f(x)=3(x+2)^2-5 \):

• Start by plotting the vertex at \((-2, -5)\).
• Draw the vertical line \( x = -2 \) to represent the axis of symmetry.
• Consider that \( a = 3 \), meaning the parabola opens upwards and is fairly narrow.
• Use symmetry to plot other key points. For example, from \((-1, -2)\) find \((-3, -2)\).

By following these steps, you create a more accurate and visually clear representation of the parabola, making it easier to understand its behavior.