/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 68 Replace \(x\) with 0 and find th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 68

Replace \(x\) with 0 and find the \(y\) -intercept for the graph of \(y=x^{2}-2 x-3\)

Short Answer

Expert verified
The y-intercept of the function \(y=x^{2}-2x-3\) is -3.

Step by step solution

01

Identify the given function

In this case, the given function is \(y=x^{2}-2x-3\).
02

Set x to 0

In order to find the y-intercept, \(x\) must be set to 0 in the equation. This will result in the equation \(y=(0)^{2}-2(0)-3\).
03

Solve for y

The calculations made when replacing \(x\) with 0, simplify the equation to \(y=-3\). So, when \(x=0\), \(y=-3\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Quadratic Function
A quadratic function is a type of polynomial that is characterized by the highest power of its variable being two. It is commonly written in the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). This equation represents a parabola when graphed on a coordinate plane. The direction of the parabola, opening upwards or downwards, is determined by the sign of \(a\). If \(a\) is positive, the parabola opens upwards. Conversely, if \(a\) is negative, it opens downwards.

Quadratic functions display a distinct set of features:
  • The vertex, which is the highest or lowest point depending on the direction the parabola opens.
  • The axis of symmetry, a vertical line that passes through the vertex and divides the parabola into two mirror-image halves.
  • The y-intercept, where the graph intersects the y-axis, found by evaluating the function at \(x = 0\).
Understanding these aspects aids in sketching the graph and analyzing the function's behavior.
Graphing
Graphing a quadratic function like \(y = x^2 - 2x - 3\) involves plotting its curve on a coordinate plane. This process provides a visual representation that helps in understanding the function's properties. Begin by identifying key features:

  • Y-intercept: As found using the original exercise, this is \(-3\), where the graph touches the y-axis at \((0, -3)\).
  • Vertex: Use the formula \(x = -\frac{b}{2a}\) to find the x-coordinate of the vertex. Here, \(a = 1\) and \(b = -2\), yielding \(x = 1\). Substitute \(x = 1\) back into the function to determine the y-coordinate of the vertex.
  • Symmetry: The graph is symmetric about the vertical line \(x = 1\), which is the axis of symmetry.
By plotting these components on the graph and joining them to form a parabola, we visualize the quadratic function. Locating its vertex, y-intercept, and symmetry allows for accurate plotting, capturing the curve's shape and direction.
Algebraic Equation
An algebraic equation is a mathematical statement that shows the equality between two expressions. In a quadratic function like \(y = x^2 - 2x - 3\), we encounter an algebraic equation of the second degree. Solving these equations involves finding the values of \(x\) that satisfy the equation, known as the roots or solutions.

When working with a quadratic equation, it can sometimes be solved by:
  • Factoring: Expressing the equation as a product of two binomials if possible.
  • Quadratic formula: Using \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which is derived from completing the square.
  • Completing the square: Rewriting the equation in vertex form by adding and subtracting terms within it.
These methods are essential for different scenarios, such as when factorization is challenging or when an exact solution is needed. Applying these algebraic techniques helps in analyzing and interpreting the characteristics of quadratic functions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use a graphing utility to solve \((x-1)^{2}-9=0\) Graph \(y=(x-1)^{2}-9\) in a \([-5,5,1]\) by \([-9,3,1]\) viewing rectangle. The equation's solutions are the graph's \(x\) -intercepts. Check by substitution in the given equation.

$$\begin{array}{l}\text { I simplified } \frac{3+2 \sqrt{3}}{2} \text { to } 3+\sqrt{3} \text { because } 2 \text { is a factor of } \\\2 \sqrt{3} \text { . }\end{array}$$., Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

If 4 times a number is decreased by 3 and this difference is squared, the result is \(9 .\) Find the number \((s)\)

Use a graphing utility to solve \(4-(x+1)^{2}=0 .\) Graph \(y=4-(x+1)^{2}\) in a \([-5,5,1]\) by \([-5,5,1]\) viewing rectangle. The equation's solutions are the graph's \(x\) -intercepts. Check by substitution in the given equation.

Solve each equation or system of equations. $$\left\\{\begin{array}{l}5 x-3 y=-13 \\ x=2-4 y\end{array}\right.$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.