91Ó°ÊÓ

When you transform a function, you essentially change its shape or position on a graph. For quadratic functions like our example, these transformations can involve sliding the graph up, down, left, or right, and even reshaping it. Let's consider the specific transformations in our exercise.

• Shift Right and Down: In the function \( y = f(x-0.5)-0.6 \), the term \( x-0.5 \) indicates a shift to the right by 0.5 units. This is because subtracting from \( x \) moves the graph to the right. Then, \(-0.6\) means moving the graph downwards by 0.6 units.
• Horizontal Compression: For \( y = f(1.5x) \), the replacement of \( x \) with \( 1.5x \) compresses the graph horizontally. Think of it as squeezing the graph towards the \( y \)-axis, making it narrower. This happens because you're multiplying \( x \) by a number greater than 1.

Understanding how these transformations affect the graph can help in predicting what the graph will look like before plotting it.

Graphing a parabola can be straightforward once you grasp the basic shape of quadratic functions. The simplest parabolic function is \( y = x^2 \), which has a distinct U-shape opening upwards. In our scenario, the function \( f(x) = x^2 - 3x \) possesses a similar shape but is adjusted slightly based on its equation.

• Vertex Calculation: The vertex of a parabola \( ax^2 + bx + c \) occurs at the point \( x = -\frac{b}{2a} \). For \( f(x) = x^2 - 3x \), which equates to \( a = 1 \) and \( b = -3 \), the vertex is found at \( x = \frac{3}{2} \).
• Opening Direction: Since \( a = 1 \) (positive), our parabola opens upwards. If \( a \) were negative, the parabola would open downwards.

Plotting this graph between \( [-2, 5] \), we find the vertex and draw our parabola, ensuring it mirrors the shape predicted by these characteristics.

Understanding the domain and range is essential for graphing functions. The domain of a function is the set of all possible input values (\( x \)-values). In our exercise, the domain is specified as \([-2, 5]\). This means we only consider \( x \)-values within this interval for our graphs.

The range, on the other hand, is the set of all possible output values (\( y \)-values) that the function can produce. For a quadratic function like \( f(x) = x^2 - 3x \), the range depends on its vertex and the direction it opens. Since our parabola opens upwards, the lowest point, or minimum \( y \)-value, will be at its vertex.

• Finding the Range: Calculate \( y \) at the vertex and that's the minimum value. Given our vertex at \( x = \frac{3}{2} \), compute \( f\left(\frac{3}{2}\right) \) to find this minimum \( y \)-value.
• Determining Full Range: As \( x \) moves within \([-2, 5]\), \( y \) will increase from this minimum point upward.

It's important in graphing to depict these limits visibly so you can understand how the graph behaves near the edges of its domain.