In this Grade 9 lesson from California Reveal Math Algebra 1, students learn how to identify x-intercepts and y-intercepts from graphs and tables, determine intervals where a function is positive or negative, and interpret the meaning of roots and zeros in real-world contexts. Students practice finding intercepts using visual graphs, data tables, and applied scenarios such as a ball's height over time or a declining account balance. The lesson builds foundational skills for solving equations graphically and analyzing function behavior across Unit 2: Relations and Functions.
Section 1
Identifying X- and Y-Intercepts
The x-intercept is the point where a graph crosses the x-axis. The y-coordinate of this point is always , so it has the form .
The y-intercept is the point where a graph crosses the y-axis. The x-coordinate of this point is always , so it has the form .
Section 2
Positive and Negative Intervals of a Function
A function has positive intervals where (graph is above the x-axis) and negative intervals where (graph is below the x-axis). At x-intercepts, and the function changes sign.
Section 3
Interpreting Intercepts in Real-World Context
In real-world application problems, the - and -intercepts represent the value of one quantity when the other quantity is zero.
An equation for a fundraiser is , where is student tickets and is adult tickets.
An equation for a school sports event is , where is the number of boys participating and is the number of girls participating.
Intercepts tell a story of extremes! The -intercept reveals what happens when you have “none” of the item on the -axis, while the -intercept shows what occurs when you have “none” of the item on the -axis. It's a great way to understand the limits of a situation.
Section 4
Zeros, Roots, and X-Intercepts Connection
Zero of a Function
For any function , if , then is a zero of the function.
Expand to review the lesson summary and core properties.
Section 1
Identifying X- and Y-Intercepts
The x-intercept is the point where a graph crosses the x-axis. The y-coordinate of this point is always , so it has the form .
The y-intercept is the point where a graph crosses the y-axis. The x-coordinate of this point is always , so it has the form .
Section 2
Positive and Negative Intervals of a Function
A function has positive intervals where (graph is above the x-axis) and negative intervals where (graph is below the x-axis). At x-intercepts, and the function changes sign.
Section 3
Interpreting Intercepts in Real-World Context
In real-world application problems, the - and -intercepts represent the value of one quantity when the other quantity is zero.
An equation for a fundraiser is , where is student tickets and is adult tickets.
An equation for a school sports event is , where is the number of boys participating and is the number of girls participating.
Intercepts tell a story of extremes! The -intercept reveals what happens when you have “none” of the item on the -axis, while the -intercept shows what occurs when you have “none” of the item on the -axis. It's a great way to understand the limits of a situation.
Section 4
Zeros, Roots, and X-Intercepts Connection
Zero of a Function
For any function , if , then is a zero of the function.