Problem 29

Question

Find the $x$ - and $y$ -intercepts of the graph of the equation. $y=x^{2}+x-2$

Step-by-Step Solution

Verified

Answer

The x-intercepts of the equation are -2 and 1, and the y-intercept is -2.

1Step 1: Finding the x-intercepts

Set y to 0 and solve the equation for x:\n\n0 = $x^{2}+x-2$\n\nFactor the quadratic equation:\n\n0 = $(x-1)(x+2)$\n\nSet each factor equal to zero and solve for x to get the x-intercepts:\n\n$x-1 = 0$ gives $x = 1$ and $x + 2 = 0$ gives $x = -2$. Therefore, the x-intercepts are -2 and 1.

2Step 2: Finding the y-intercept

Set x to 0 and solve the equation for y:\n\n$y = 0^{2} + 0 - 2$\n\nThis simplifies to $y = -2$. Therefore, the y-intercept is -2.

Key Concepts

Quadratic EquationsFactoringCoordinate PlaneIntercepts Calculation

Quadratic Equations

Quadratic equations are fundamental algebraic expressions found in the form of $ ax^2 + bx + c = 0 $. They include a variable squared, ensuring the highest exponent of the variable is two. This is what makes them "quadratic." In our equation, $ y = x^2 + x - 2 $, it's clear that the variable $ x $ is raised to the power of 2, aligning with the standard form of quadratic equations. Quadratics can describe the path of a thrown ball or the shape of a satellite dish. It's important to identify the coefficients, which are the numbers in front of each term: $ a = 1 $, $ b = 1 $, and $ c = -2 $. This isn’t just theoretical; these numbers define the curve's width, direction, and position on the graph. Whether it's finding vertex points or solving for intercepts, understanding the structure of quadratic equations is crucial.

Factoring

Factoring is a method used to simplify quadratic equations to find their roots or intercepts easily. Essentially, it's breaking down the equation into simpler expressions that, when multiplied together, give the original quadratic. Think of it like splitting a number into its multiplication components. For the equation $ y = x^2 + x - 2 $, factoring turns it into $ (x-1)(x+2) = 0 $. This transformation helps us solve for the values of $ x $ when $ y = 0 $.

Factorization makes it easier to see solutions.
Simplified equations reveal critical points on a graph.

By setting each factor equal to zero, we find the solutions $ x = 1 $ and $ x = -2 $, meaning these are the points where the graph crosses the $ x $-axis (x-intercepts). Factoring is like finding a shortcut to linear solutions hidden in quadratic complexity.

Coordinate Plane

The coordinate plane, or Cartesian coordinate system, is a two-dimensional surface defined by two axes: horizontal $ x $-axis and vertical $ y $-axis. Every point on this plane is represented by a pair $(x, y)$. In graphing equations like $ y = x^2 + x - 2 $, each solution or intercept is a point on this plane.

The $ x $-axis is a horizontal line.
The $ y $-axis is a vertical line.

Understanding the coordinate plane is vital, as it allows us to visualize where the graph of an equation like a quadratic lies. Each intercept corresponds to a point of intersection between the curve and one of the axes. Navigating this plane involves understanding these axes' alignment and how each point corresponds to the equation's solution. By plotting these intercepts, we get a graphical representation of the equation's solutions on the coordinate plane.

Intercepts Calculation

Calculating intercepts is all about finding where a curve crosses the axes. In a quadratic equation, the $ x $-intercepts occur where $ y = 0 $, and the $ y $-intercept is found where $ x = 0 $.

Finding $ x $-Intercepts

The solution process involves setting $ y = 0 $ in the equation and solving for $ x $. For our equation $ y = x^2 + x - 2 $, it simplifies to $ 0 = (x-1)(x+2) $. Solving these factors gives:

$ x = 1 $
$ x = -2 $

Finding $ y $-Intercept

To find the $ y $-intercept, substitute $ x = 0 $ into the equation. This gives $ y = -2 $, meaning the graph crosses the $ y $-axis at $ (0, -2) $.

Key Takeaways

$ x $-intercepts are points where the graph touches the $ x $-axis.
The $ y $-intercept is where the graph touches the $ y $-axis.

Understanding intercepts helps in sketching the graph accurately. It also aids in interpreting the graph's behavior and the quadratic function's nature.

Problem 29

Problem 30