91Ó°ÊÓ

A quadratic function in vertex form is especially helpful when sketching graphs.
It allows you to easily identify the key features of the function. The vertex form of a quadratic is given by \[ f(x) = a(x-h)^2 + k \] where:

• \( a \): a constant that affects the width and direction of the parabola.
• \( h \): the x-coordinate of the vertex.
• \( k \): the y-coordinate of the vertex.

By looking at the equation in vertex form, you can directly see the vertex of the parabola as \((h, k)\). This feature makes it simpler to sketch the graph because you know exactly where the vertex is located on the coordinate plane.
In our exercise, the function \(3(x-4)^2+1\) shows the vertex at \((4, 1)\). This gives you a starting point for your graph.

Parabolas are a type of curve that are symmetric and have a characteristic U-shape.
In the case of quadratic functions, the graph of the function is always a parabola. Depending on the value of \( a \) in the equation, the parabola can either open upwards or downwards:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

For our specific quadratic function \(3(x-4)^2+1\), since \( a = 3 \) (a positive number), our parabola opens upwards.

Besides the direction, \( a \) also affects how wide the parabola opens. When \( \vert a \vert > 1 \), the parabola becomes narrower than the standard \( x^2 \) parabola. In contrast, when \( \vert a \vert < 1 \), the parabola becomes wider.
This means our parabola, with \( a = 3 \), will be narrower than the usual \( x^2 \) graph.

Understanding transformations of the base function \( f(x) = x^2 \) is crucial for graphing more complex quadratic functions.
Transformations include shifting, stretching, or reflecting the graph. For our equation \( f(x) = 3(x-4)^2 + 1 \), transformations play a key role:

• **Horizontal Shift**: The expression \((x-4)\) means the graph shifts 4 units to the right.
• **Vertical Shift**: The \(+1\) at the end of the equation indicates a shift 1 unit upwards.
• **Vertical Stretch**: The coefficient \(3\) makes the parabola narrower compared to the regular parabola \( x^2 \), stretching it vertically.

These transformations allow you to take a simple parabola and move or stretch it to fit different requirements.
They are easy to apply when the parabola is in vertex form, enabling quick and efficient graphing.