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Chapter 3: Problem 97

Write the equation of each parabola in standard form. Vertex: \((-3,-4) ;\) The graph passes through the point \((1,4)\)

Short Answer

Expert verified
The equation of the parabola in standard form is \(y = \frac{2}{5}(x + 3)^2 - 4\).

Step by step solution

01

Input the vertex into the equation

First, we need to insert the vertex into the standard equation. Our vertex is (-3,-4), so h is -3 and k is -4. Therefore, after inserting, our equation is \(y = a(x - (-3))^2 - 4\), which simplifies to \(y = a(x + 3)^2 - 4\).
02

Find 'a' by using the given point

The given point that the graph passes through is (1,4). We will substitute these values into the equation to solve for 'a'. Inserting x=1 and y=4, we get: \(4 = a(1 + 3)^2 - 4\). Simplifying and solving the equation gives: \( a= \frac{2}{5}\).
03

Write the final equation in standard form

Now that we have a value for 'a' as \(\frac{2}{5}\), We substitute this value back to the equation we got in step 1 to get the final equation of the parabola: \(y = \frac{2}{5}(x + 3)^2 - 4\).

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Most popular questions from this chapter

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=x^{3}(x+2)^{2}(x+1)$$

In Exercises \(98-99,\) use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the \([\mathrm{ZOOMOUT}]\) feature to show that \(f\) and \(g\) have identical end behavior. $$f(x)=-x^{4}+2 x^{3}-6 x, \quad g(x)=-x^{4}$$

What is a rational function?

Is every rational function a polynomial function? Why or why not? Does a true statement result if the two adjectives rational and polynomial are reversed? Explain.

Use long division to rewrite the equation for \(g\) in the form $$ \text {quotient}+\frac{\text {remainder}}{\text {divisor}} $$ Then use this form of the function's equation and transformations \( \text { of } f(x)=\frac{1}{x} \text { to graph } g \). $$ g(x)=\frac{3 x-7}{x-2} $$
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