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Quadratic functions can be written in a special format called the vertex form, which makes it easier to identify the vertex of the parabola. The vertex form is given by:
\( y = a(x - h)^2 + k \).
- Here, \(a\) is the coefficient that stretches or compresses the parabola.
- \(h\) and \(k\) are the coordinates of the vertex of the parabola.
For example, take part (a) of the exercise:
The given values are \(a = 1, h = 2, k = -4\).
The vertex form becomes:
\(y = 1(x - 2)^2 - 4 \).
From this equation, you can see that the parabola has a vertex at \( (2, -4) \). It opens upwards because the coefficient \((a = 1)\) is positive.

Another way to express quadratic functions is the standard form: \( y = ax^2 + bx + c \).
Here, \(a, b,\) and \(c\) are constants. This form is useful for identifying the y-intercept (\b{c}\b) and for some algebraic manipulations.
To convert from vertex form to standard form, you expand and simplify the equation.
Using part (b)'s example, with \(a = -1, h = 4, k = 3 \):
Vertex form: \( y = -1(x - 4)^2 + 3 \)
First, expand \( (x - 4)^2 \rightarrow x^2 - 8x + 16 \).
Then, the equation becomes: \( y = -1(x^2 - 8x + 16) + 3 \).
Distribute the -1: \( y = -x^2 + 8x - 16 + 3 \).
Combine like terms to get: \( y = -x^2 + 8x - 13 \).
Now, you have the standard form: \( y = -x^2 + 8x - 13 \).

Graphing quadratic functions helps visualize their properties, like the vertex, axis of symmetry, and direction of opening.

• Vertex: In the vertex form, \((h, k)\) is the vertex of the parabola.
• Axis of Symmetry: It is the vertical line that passes through the vertex, given by \( x = h \).
• Direction: If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards.

For instance, part (c) with \( a = -2, h = -3, k = 1 \):
The equation in vertex form is: \( y = -2(x + 3)^2 + 1 \).
It converts to standard form as: \( y = -2x^2 - 12x - 17 \).
On the graph:
- The vertex is at \(( -3, 1 )\).
- It opens downwards because \( a = -2\) is negative.
- The axis of symmetry is \( x = -3 \).
These aspects help in correctly drawing the graph. Utilizing technology and graphing programs can further verify accuracy.