Question

Use shifts and scalings to graph then 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. $$p(x)=x^{2}+3 x-5$$

Short answer

Question: Apply shifts and scalings to the function $$p(x)=x^{2}+3 x-5$$, graph it, and then verify your work using a graphing utility. Answer: Using the steps provided, we have applied the necessary shifts and scalings on the given function $$p(x)=x^{2}+3 x-5$$ and graphed it as a parabola with its vertex, axis of symmetry, and x-intercepts. To verify the correctness of the graph, use a graphing utility, such as Desmos or a graphing calculator, and compare the result to the graph created in Step 4.

Step-by-step solution

Identify the original function

The given function is a quadratic function of the form $$p(x)=x^2+3x-5$$. We can consider the original function as: $$f(x)=x^2$$ Now, we will perform shifts and scalings on this original function to obtain the given function $$p(x)$$.

Apply horizontal shift

The given function has a term $$3x$$. To include this term in our original function, we need to apply a horizontal shift. We can rewrite the given function as: $$p(x)=x^2+(3x)$$

Apply vertical shift

Next, we need to apply a vertical shift to include the constant term "-5" in our function. We can now rewrite our function as: $$p(x)=x^2+(3x)-5$$

Graph the resulting function

Now that we have applied both horizontal and vertical shifts and obtained the given function $$p(x)$$, we can graph it. To graph the function, plot the parabola of the equation $$p(x) = x^2 + 3x - 5$$ along with its vertex, axis of symmetry, and x-intercepts.

Verify using a graphing utility

Use a graphing utility, such as Desmos or a graphing calculator, to plot the function $$p(x) = x^2 + 3x - 5$$. Compare the graph from the graphing utility to the one created in Step 4 to ensure the shifts and scalings are correct, and the graph correctly represents the given function.

Key concepts

Graphing Transformations

Graphing transformations are changes that we apply to the graph of a function to produce a new function. These can include shifts (translations), reflections, and scalings (stretches and compressions).
For the quadratic function presented, transformations were employed to transition from an original function like \( f(x)=x^2 \) to our target function \( p(x)=x^2+3x-5 \). Here are the specific transformations used:

• Horizontal Shifts: These involve moving the graph left or right along the x-axis. For the given function, the \( 3x \) term suggests an operation similar to completing the square, which results in a shift in the graph.
• Vertical Shifts: This moves the graph up or down along the y-axis. For our quadratic function, the "-5" represents a vertical shift downwards by 5 units.

All these shifts reposition the vertex and alter the parabola's original placement on the coordinate plane.

Vertex Form

The vertex form of a quadratic equation is extremely useful for quickly identifying the vertex of the parabola represented by the quadratic function. It's given by the expression \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex.
To convert your standard quadratic form \( ax^2 + bx + c \) into vertex form, you will often "complete the square". For the function \( p(x) = x^2 + 3x - 5 \), while it is not rewritten in vertex form, completing the square internally helps us graph it effectively.
This method allows understanding:

• Where the parabola's vertex is located.
• The direction in which the parabola opens (upward or downward).
• Whether the parabola has been vertically stretched or compressed (determined by the value \( a \)).

Understanding vertex form not only simplifies graphing but also aids in connecting transformations back to the original function.

Graphing Utilities

Graphing utilities, like Desmos or graphing calculators, are powerful tools for visualizing mathematical functions. They are especially helpful when working with more complex functions or checking manual graphs for accuracy. For the quadratic function \( p(x) = x^2 + 3x - 5 \), you can input the function into a graphing utility to see the parabola in a clear visual form.
These tools help you:

• Explore the effects of transformations more dynamically by seeing changes in real-time as you adjust parameters.
• Confirm calculations by providing visual evidence that matches or corrects manual graph work.
• Experiment with various functions and understand their behavior without doing manual calculations every time.

Using graphing utilities makes learning about quadratic functions and their transformations an interactive and more engaging process.