Question

Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed. $$h(x)=-4 x^{2}-4 x+12$$

Short answer

Answer: The vertex of the given quadratic function is at the point $$\left(-\frac{1}{2}, 7\right)$$. The parabola opens downwards since the coefficient of the x² term is negative.

Step-by-step solution

Identify the original function and vertex form

We are given a quadratic function and we need to identify an original function on which the shifts and scalings are performed. The standard form of a quadratic function is: $$f(x)=ax^2+bx+c$$ We are given: $$h(x)=-4x^2-4x+12$$ The vertex form of a quadratic function is given by: $$g(x)=a(x-h)^2+k$$ We will form the vertex form of the given function, find the vertex, and then apply shifts and scalings to the graph.

Obtain the vertex form

In order to convert the given function into vertex form, we need to complete the square. We will first factor out the coefficient of the x^2 term: $$h(x)=-4(x^2+x-3)$$ Now, we complete the square inside the parentheses by adding and subtracting the square of half of the coefficient of the linear term: $$h(x)=-4\left(x^2+x+\left(\frac{1}{2}\right)^2-3-\left(\frac{1}{2}\right)^2\right)$$ We obtain the vertex form: $$h(x)=-4\left(x+\frac{1}{2}\right)^2+7$$

Determine the vertex, and any symmetry properties of the function

From the vertex form of the function, we can determine that the vertex is located at: $$(h,k)=\left(-\frac{1}{2}, 7\right)$$ Since the coefficient of the x^2 term is negative, the parabola will open downwards.

Apply the shifts and scalings, sketch the graph

With the given vertex form, we can apply the following shifts and scalings to the original function: 1. Horizontally shift the graph to the left by \(1/2\) units (\(h=-1/2\)). 2. Vertically scale the graph by a factor of \(-4\) (negative indicating a reflection across the x-axis). 3. Vertically shift the graph up by \(7\) units (\(k=7\)). These transformations will result in the given function, and a graph of the function can be sketched.

Check the graph using a graphing utility

Make use of a graphing utility, such as Desmos, to confirm that the modifications performed on the original function result in the correct graph. You should find that the graph matches the transformations we applied in Step 4.

Key concepts

Vertex Form

When working with quadratic functions, the vertex form is a particularly useful tool. It allows us to easily identify the vertex of the parabola, and it sets the stage for understanding how we can transform the graph through shifts and scalings. A quadratic function in vertex form looks like this: \[ g(x) = a(x-h)^2 + k \]

• \( a \) determines the width and the direction of the parabola.
• \( h \) determines the horizontal shift.
• \( k \) denotes the vertical shift.

In order to convert a given quadratic from standard form \( f(x) = ax^2 + bx + c \) to vertex form, you will need to complete the square. This involves rearranging and adjusting the function so that it fits the format of vertex form. Completing the square allows you to tidy up expressions and show clearly any transformations the graph will undergo.Remember, the vertex form reveals both the vertex \((h, k)\) and the symmetry properties of the parabola, such as whether it opens upwards or downwards. Understanding these basics simplifies analyzing and transforming quadratic functions.

Shifts and Scalings

Transformations of a quadratic function involve shifts and scalings, and they are an essential part of graphing functions accurately. Once we have a function in vertex form, \( g(x) = a(x-h)^2 + k \), we can easily see how the graph is transformed.

• Horizontal Shifts: If \( h \) is positive, the graph shifts to the right. If \( h \) is negative, it shifts to the left.
• Vertical Shifts: The value of \( k \) shifts the graph vertically. A positive \( k \) moves it upwards, while a negative \( k \) moves it downwards.
• Vertical Scaling: The value of \( a \) affects the steepness and direction of the parabola. If \( a \) is greater than 1, the graph stretches vertically, making it narrower. If \( 0 < a < 1 \), it compresses, making the graph wider. Negative values of \( a \) reflect the graph across the x-axis, inverting the direction of opening.

By understanding these shifts and scalings, you can transform the quadratic function to align with any given parameters, helping you identify shifts from an ‘original’ or parent form of \( y = x^2 \). This is a foundational skill in graph analysis, allowing modifications that aid in clear visualization of function behavior in real-world applications.

Graphing Utilities

Graphing utilities, like Desmos or a graphing calculator, are invaluable tools when working with quadratic functions. These utilities provide a visual representation of the equations, making it easier to verify the transformations, such as shifts and scalings, that we calculate manually.

• Plotting your function using a graphing utility lets you quickly see the shape and position of the parabola.
• You can compare the manual drawing with the plotted graph to see the effect of transformations like horizontal and vertical shifts.
• These tools are interactive, allowing you to adjust the coefficients \( a \), \( h \), and \( k \) to see real-time changes in the graph.

By using graphing utilities, not only can you confirm your manual transformations, but you also gain deeper insights into the behavior of quadratic functions by experimenting with different values. This interactive approach can significantly enhance understanding, making complex concepts more tangible and accessible.