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Chapter 2: Problem 51

Find a function \(y=a x^{2}+b x+c\) whose graph has an \(x\) -intercept of \(1,\) a \(y\) -intercept of \(-2,\) and a tangent line with a slope of \(-1\) at the \(y\) -intercept.

Short Answer

Expert verified
The function is given by \( y = 3x^2 - x - 2 \).

Step by step solution

01

Analyze the x-intercept

The function has an x-intercept at \( x = 1 \). This means that the function passes through the point \( (1, 0) \). Substituting this into the function gives the equation:\[a(1)^2 + b(1) + c = 0 \]which simplifies to:\[a + b + c = 0\]
02

Analyze the y-intercept

The function has a y-intercept at \( y = -2 \). This means that the function passes through the point \( (0, -2) \). Substituting this into the function gives the equation:\[c = -2\]
03

Determine the slope condition

At the y-intercept, the slope of the tangent line is \(-1\). The derivative \( y' = 2ax + b \) gives the slope. We need to find the slope when \( x = 0 \):\[y'(0) = 2a(0) + b = b = -1\]Thus, we have \( b = -1 \).
04

Solve the system of equations

Now we have the following equations:1. \( a + b + c = 0 \)2. \( c = -2 \)3. \( b = -1 \)Substitute \( c = -2 \) and \( b = -1 \) into the first equation:\[a - 1 - 2 = 0\]\[a - 3 = 0\]Thus, \( a = 3 \).
05

Write the final function

With \( a = 3 \), \( b = -1 \), and \( c = -2 \), the function becomes:\[y = 3x^2 - x - 2\]

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Key Concepts

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

Understanding x-intercepts in Quadratic Functions
When tackling quadratic functions, understanding x-intercepts is crucial. An x-intercept of a function refers to a point where the graph of the function crosses the x-axis.
This means that at the x-intercept, the output value ( y ) of the function is 0.
  • For the quadratic function y = ax^2 + bx + c , finding the x-intercepts requires solving the equation ax^2 + bx + c = 0 .
  • This can be done using methods such as factoring, completing the square, or utilizing the quadratic formula.
  • In the original problem, given an x-intercept at x = 1 , substituting this value into the equation gives us important information to determine coefficients a , b , and c .
An x-intercept not only defines a key point on the graph, but it also aids in constructing the entire quadratic equation.
Decoding y-intercepts in a Quadratic Graph
A y-intercept, conversely, is the point where the graph crosses the y-axis. At this point, the x-coordinate is always 0.
The role of the y-intercept is to reveal where the function reaches a specified height at x = 0.
  • For a quadratic function y = ax^2 + bx + c , the y-intercept is straightforwardly found when x = 0 .
  • This simplifies the equation to y = c , indicating that c is the y-intercept.
  • In our original problem, the y-intercept is given as -2 , telling us directly that c = -2 .
Understanding y-intercepts helps in quickly placing the quadratic curve on the graph and determining other characteristics by giving a reference point from which the graph extends.
Exploring Tangent Lines to Quadratics
In the world of quadratic functions, tangent lines play an essential role. A tangent line touches the quadratic curve at exactly one point, and its slope can tell us a lot about the function's behavior at that point.
This slope can be derived using calculus.
  • The derivative of the quadratic function y = ax^2 + bx + c is given by y' = 2ax + b .
  • This derivative represents the slope of the tangent line at any given point x on the function.
  • For the problem we have, the slope of the tangent line at the y-intercept (0, -2) is -1 , which directly leads us to find the coefficient b when substituted into the derivative formula as 0 for x .
Incorporating the tangent line's properties into our quadratic function provides further insight, ensuring that the graph aligns with expected mathematical behavior at specified points.

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