Algebra

a) Complete the table of values for y = 2x + 3 x | -2 | -1 | 0 | 1 | 2 | 3 y | | 1 | | | 7 | b) Draw the graph of y = 2x + 3 on the grid.

Step-by-step solution with explanation

Final Answer

Completed table: x = -2 → y = -1, x = -1 → y = 1, x = 0 → y = 3, x = 1 → y = 5, x = 2 → y = 7, x = 3 → y = 9. For the graph, plot all six coordinate pairs and join them with a straight line using a ruler.

Step-by-step solution

Understand the rule y = 2x + 3

For every x value, multiply it by 2 then add 3. This gives us the matching y value. We already know x = -1 gives y = 1, and x = 2 gives y = 7 — let's verify and fill the rest.

Calculate y when x = -2

Plug x = -2 into the formula. Multiply 2 × -2 = -4, then add 3 to get -1.

Verify y when x = -1

✓

The table already shows y = 1 for x = -1. Our formula confirms this is correct.

Calculate y when x = 0

Plug x = 0 into the formula. Anything times 0 is 0, so y = 3. This is also the y-intercept of the line.

Calculate y when x = 1

Plug x = 1 into the formula. Multiply 2 × 1 = 2, then add 3 to get 5.

Verify y when x = 2, find y at x = 3

✓ and

The table already shows y = 7 for x = 2 — confirmed. For x = 3, multiply 2 × 3 = 6, then add 3 to get 9.

Write the completed table of values

All six y values are now filled in. Notice the y values go up by 2 each time x goes up by 1 — that's the slope of 2 in action.

Describe how to draw the graph

Plot each (x, y) pair from the table on a grid. Because this is a linear equation (no x² or higher), all points lie on one perfectly straight line. Use a ruler to connect them and extend the line in both directions.

Understanding this problem

Learning Insight

The equation y = 2x + 3 is in slope-intercept form (y = mx + b). The number 2 is the slope — it tells you the line rises 2 units for every 1 unit you move right. The number 3 is the y-intercept — where the line crosses the y-axis. Understanding these two numbers lets you sketch any straight-line graph quickly without a full table.

Quick Tip

Once you know any two points, you can draw the whole line — you don't need all six! The easiest two to find fast are the y-intercept (set x = 0) and the x-intercept (set y = 0). For this line, x-intercept: 0 = 2x + 3, so x = -1.5.

Common Mistake

Students often forget to multiply BEFORE adding when substituting negatives. For example with x = -2, they write 2 + (-2) + 3 instead of 2×(-2) + 3, getting the wrong sign and a wrong y value.