Answer :
To understand how the graph, equation, and table of values of a function are interconnected, let's examine the relationship using the linear function \( y = 2x + 5 \). We'll break it down into separate parts: the equation, the table of values, and the graph.
### 1. The Equation
The equation \( y = 2x + 5 \) represents a linear relationship between \( x \) and \( y \). For any given value of \( x \), you can find the corresponding value of \( y \) by plugging \( x \) into the equation.
### 2. The Table of Values
A table of values presents specific pairs of \( x \) and \( y \) that satisfy the equation \( y = 2x + 5 \):
[tex]\[ \begin{tabular}{|c|c|} \hline x & y \\ \hline -2 & 1 \\ \hline 0 & 5 \\ \hline 2 & 9 \\ \hline 4 & 13 \\ \hline 6 & 17 \\ \hline 8 & 21 \\ \hline \end{tabular} \][/tex]
Let's verify that these values satisfy the equation \( y = 2x + 5 \):
- For \( x = -2 \):
[tex]\[ y = 2(-2) + 5 = -4 + 5 = 1 \][/tex]
- For \( x = 0 \):
[tex]\[ y = 2(0) + 5 = 0 + 5 = 5 \][/tex]
- For \( x = 2 \):
[tex]\[ y = 2(2) + 5 = 4 + 5 = 9 \][/tex]
- For \( x = 4 \):
[tex]\[ y = 2(4) + 5 = 8 + 5 = 13 \][/tex]
- For \( x = 6 \):
[tex]\[ y = 2(6) + 5 = 12 + 5 = 17 \][/tex]
- For \( x = 8 \):
[tex]\[ y = 2(8) + 5 = 16 + 5 = 21 \][/tex]
All these pairs of \( x \) and \( y \) values satisfy the equation \( y = 2x + 5 \).
### 3. The Graph
The graph is a visual representation of the function \( y = 2x + 5 \) on a coordinate plane. By plotting the pairs \((-2, 1)\), \((0, 5)\), \((2, 9)\), \((4, 13)\), \((6, 17)\), and \((8, 21)\) from the table onto the coordinate plane, we can draw the graph of the function \( y = 2x + 5 \).
Here's how the points would look:
- Point: \((-2, 1)\)
- Point: \((0, 5)\)
- Point: \((2, 9)\)
- Point: \((4, 13)\)
- Point: \((6, 17)\)
- Point: \((8, 21)\)
When you plot these points on a graph, they should fall on a straight line because the equation \( y = 2x + 5 \) is a linear equation. The line will have a slope of 2 (since for every 1 unit increase in \( x \), \( y \) increases by 2 units) and a y-intercept of 5 (the point where the line crosses the \( y \)-axis).
### Relationship Conclusion
- Equation: Provides the general formula that relates \( x \) and \( y \).
- Table: Offers specific pairs of values that satisfy the equation.
- Graph: Visually represents the relationship by plotting the points from the table on a coordinate plane and showing the line that corresponds to the equation.
All three methods describe the same function and convey the same information in different forms. The equation gives the rule, the table provides concrete examples, and the graph shows it visually.
### 1. The Equation
The equation \( y = 2x + 5 \) represents a linear relationship between \( x \) and \( y \). For any given value of \( x \), you can find the corresponding value of \( y \) by plugging \( x \) into the equation.
### 2. The Table of Values
A table of values presents specific pairs of \( x \) and \( y \) that satisfy the equation \( y = 2x + 5 \):
[tex]\[ \begin{tabular}{|c|c|} \hline x & y \\ \hline -2 & 1 \\ \hline 0 & 5 \\ \hline 2 & 9 \\ \hline 4 & 13 \\ \hline 6 & 17 \\ \hline 8 & 21 \\ \hline \end{tabular} \][/tex]
Let's verify that these values satisfy the equation \( y = 2x + 5 \):
- For \( x = -2 \):
[tex]\[ y = 2(-2) + 5 = -4 + 5 = 1 \][/tex]
- For \( x = 0 \):
[tex]\[ y = 2(0) + 5 = 0 + 5 = 5 \][/tex]
- For \( x = 2 \):
[tex]\[ y = 2(2) + 5 = 4 + 5 = 9 \][/tex]
- For \( x = 4 \):
[tex]\[ y = 2(4) + 5 = 8 + 5 = 13 \][/tex]
- For \( x = 6 \):
[tex]\[ y = 2(6) + 5 = 12 + 5 = 17 \][/tex]
- For \( x = 8 \):
[tex]\[ y = 2(8) + 5 = 16 + 5 = 21 \][/tex]
All these pairs of \( x \) and \( y \) values satisfy the equation \( y = 2x + 5 \).
### 3. The Graph
The graph is a visual representation of the function \( y = 2x + 5 \) on a coordinate plane. By plotting the pairs \((-2, 1)\), \((0, 5)\), \((2, 9)\), \((4, 13)\), \((6, 17)\), and \((8, 21)\) from the table onto the coordinate plane, we can draw the graph of the function \( y = 2x + 5 \).
Here's how the points would look:
- Point: \((-2, 1)\)
- Point: \((0, 5)\)
- Point: \((2, 9)\)
- Point: \((4, 13)\)
- Point: \((6, 17)\)
- Point: \((8, 21)\)
When you plot these points on a graph, they should fall on a straight line because the equation \( y = 2x + 5 \) is a linear equation. The line will have a slope of 2 (since for every 1 unit increase in \( x \), \( y \) increases by 2 units) and a y-intercept of 5 (the point where the line crosses the \( y \)-axis).
### Relationship Conclusion
- Equation: Provides the general formula that relates \( x \) and \( y \).
- Table: Offers specific pairs of values that satisfy the equation.
- Graph: Visually represents the relationship by plotting the points from the table on a coordinate plane and showing the line that corresponds to the equation.
All three methods describe the same function and convey the same information in different forms. The equation gives the rule, the table provides concrete examples, and the graph shows it visually.
