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Chapter 4: Problem 29

Find the rule of the quadratic function whose graph satisfies the given conditions. Vertex at (0,0)\(;\) passes through (2,12)

Short Answer

Expert verified
Answer: The rule for the quadratic function is \(y = 3x^2\).

Step by step solution

01

Write down the vertex form of the quadratic function

We are given the vertex at \((h, k) = (0, 0)\), so the quadratic function is in the form: \(y = a(x-h)^2 + k = a(x-0)^2 + 0 = ax^2\)
02

Substitute the point (2,12) into the formula

Now, we are given a point (2, 12) through which the parabola passes. We can use this to find the value of "\(a\)". Substitute \(x = 2\) and \(y = 12\) in the equation \(y = ax^2\): \(12 = a(2^2)\)
03

Solve for "a"

Now, we solve for "\(a\)" in the equation \(12 = a(2^2)\): \(12 = 4a\) Divide both sides by 4: \(a = 3\)
04

Write the final rule for the quadratic function

Now that we have the value of "\(a\)", we can write the rule for the given quadratic function: \(y = 3x^2\) So, the rule for the quadratic function whose graph satisfies the given conditions is \(y = 3x^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex Form
The vertex form of a quadratic function is an efficient way to express the equation of a parabola. The standard vertex form looks like this:
  • \( y = a(x - h)^2 + k \)
In this form, \((h, k)\) represents the vertex of the parabola, and "\(a\)" affects the width and direction of the parabola.
A positive value of \(a\) means the parabola opens upwards, while a negative value causes it to open downwards.

Knowing the vertex is incredibly helpful for quickly sketching the graph of the quadratic equation. By altering the values of \(h\) and \(k\), you can shift the parabola left, right, up, or down without changing its shape.
Parabola
A parabola is a symmetrical, U-shaped curve that represents the graph of a quadratic function. It can open upwards or downwards, depending on the sign of the "\(a\)" constant in our equation.
  • When the parabola opens upwards, it makes a smiley face.
  • If it opens downward, it's more of a frown shape.
The vertex of the parabola is the point where it either reaches its highest or lowest value, depending on its direction of opening.

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two equal halves. This line helps us understand that whatever occurs on one side of the axis will be mirrored on the other side.
Solving Quadratic Equations
Solving quadratic equations involves finding the values of \(x\) that make the equation equal to zero. There are several methods to solve these equations:
  • Factoring: Splitting the equation into simpler expressions that can be easily solved.
  • Quadratic Formula: Using the formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\) for equations ax^2 + bx + c = 0.
  • Completing the Square: Rewriting the equation in vertex form to make it easier to solve.

In this specific problem, since we started with the vertex form, it was straightforward to find "\(a\)" by substituting the given point into the equation. Once we found "\(a=3\)", the quadratic formula \(y = 3x^2\) was ready for any further evaluations of \(x\). Sometimes, just having the vertex form makes finding these solutions that much more efficient.

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Most popular questions from this chapter

A box with a square base and a volume of 1000 cubic inches is to be constructed. The material for the top and bottom of the box costs \(\$ 3\) per 100 square inches, and the material for the sides costs \(\$ 1.25\) per 100 square inches. (a) If \(x\) is the length of a side of the base, express the cost of constructing the box as a function of \(x .\) (b) If the side of the base must be at least 6 inches long, for what value of \(x\) will the cost of the box be \(\$ 7.50 ?\)

Analyze the function algebraically. List its vertical asymptotes, holes, y-intercept, and horizontal asymptote, if any. Then sketch a complete graph of the function. $$f(x)=\frac{2 x}{x+1}$$

A rectangular garden with an area of 200 square meters is to be located next to a building and fenced on three sides, with the building acting as a fence on the fourth side. (a) If the side of the garden parallel to the building has length \(x\) meters, express the amount of fencing needed as a function of \(x\). (b) For what values of \(x\) will less than 60 meters of fencing be needed? (c) What value of \(x\) will result in the least possible amount of fencing being used? What are the dimensions of the garden in this case?

Determine if \(g(x)\) is a factor of \(f(x)\) without using synthetic division or long division. $$\begin{aligned} &f(x)=(3+i) x^{3}+(1-2 i) x^{2}+(2+i) x+(1-i)\\\ &g(x)=x-i \end{aligned}$$

Determine if \(g(x)\) is a factor of \(f(x)\) without using synthetic division or long division. $$f(x)=3 x^{3}+5 x^{2}-2 x+3 ; \quad g(x)=x+1$$
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