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Chapter 1: Problem 83

What effect does increasing the constant \(c\) have on the graph of \(f(x)=a x^{2}+b x+c ?\)

Short Answer

Expert verified
Increasing the constant \(c\) in the function \(f(x) = ax^2 + bx + c\) results in a shift of the graph upwards along the y-axis.

Step by step solution

01

Define the Quadratic Function

A quadratic function is a function that can be written in the form \( f(x) = ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants with \(a 鈮� 0\). The value of \(c\) is the y-intercept, or the value of the function when \(x=0\).
02

Understand the Role of the Constant \(c\)

The parameter \(c\) is the constant term in the quadratic equation. This term represents the value of the function when \(x=0\), or the point where the graph intersects the y-axis. This is known as the y-intercept of the graph.
03

Effect of Increasing the Constant \(c\)

When we increase the Constant \(c\), the entire graph of the function will shift upwards along the y-axis. This means that the graph of the function \( f(x) = ax^2+bx+c+d \), where \(d>0\), will be above the graph of the function \( f(x) = ax^2+bx+c \).

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Key Concepts

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

Understanding the Y-intercept
In the context of quadratic functions, the y-intercept is a crucial concept. It is where the graph of the function crosses the y-axis. For a quadratic equation in the form of \( f(x) = ax^2 + bx + c \), the y-intercept is represented by the constant term \( c \).

This value can be easily identified by simply setting the value of \( x \) to zero. Therefore, the y-intercept is the point \( (0, c) \) on the graph.
  • It serves as a starting point for sketching the graph.
  • The y-intercept provides valuable information about the function's vertical position.
Knowing the y-intercept can aid in better understanding and predicting the behavior of the quadratic graph.
Exploring Graph Transformations
Graph transformations involve changing the position or shape of the graph. For quadratic functions, transformations typically include:
  • Vertical and horizontal shifts;
  • Reflections;
  • Stretching or compressing the graph.

Increasing the constant \( c \) in the quadratic function equation \( f(x) = ax^2 + bx + c \) results in a vertical shift of the graph upward. This transformation doesn't affect the shape of the parabola, only its position.
Hence, if you visualize changing \( c \), imagine moving the entire graph up or down along the y-axis based on the value of \( c \). This simple transformation can have significant impacts on how the quadratic function is interpreted and applied in various scenarios.
Delving into Quadratic Equations
Quadratic equations lie at the heart of many mathematical concepts due to their unique properties. They are usually expressed in the standard form: \( f(x) = ax^2 + bx + c \), where:
  • \( a eq 0 \) ensures the presence of the quadratic term \( x^2 \);
  • \( b \) affects the symmetry and slope of the parabola;
  • \( c \) influences the vertical position, as it represents the y-intercept.

Quadratic equations form parabolas when graphed on a Cartesian plane. The direction of these parabolas (upward or downward) is dictated by the sign of the term \( a \). Positive \( a \) values make the parabola open upwards, while negative \( a \) values direct it downward.
Understanding these elements provides a more comprehensive view of how different parameters impact the graph, allowing for greater control and prediction when working with quadratic functions.

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Most popular questions from this chapter

solve by completing the square or by using the quadratic formula. $$\frac{1}{2} x^{2}+\frac{3}{4} x-1=0$$

Use interval notation to express the solution set of each inequality. $$|2 x-5| \geq 1$$

Use interval notation to express the solution set of each inequality. $$|x+3|>30$$

Factor: \(3 x^{2}+10 x-8[\text { A. } 3]\)

The notation \(\left.f(x)\right|_{a} ^{b}\) is used to denote the difference \(f(b)-f(a) .\) That is, $$\left.f(x)\right|_{a} ^{b}=f(b)-f(a)$$ Evaluate \(\left.f(x)\right|_{0} ^{b}\) for the given function \(f\) and the indicated values of \(a\) and \(b\). $$f(x)=2 x^{3}-3 x^{2}-x ;\left.f(x)\right|_{0} ^{2}$$
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