GCSE Maths: Solving equations using graphs (Foundation)
Part of the GCSE Maths quadratic and other graphs chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Solving equations using graphs
A graph can be used to solve an equation. This is useful when an equation is hard to solve in any other way, and it shows you what the solutions mean. The idea is always the same: a solution is the x-value of a point where two graphs meet.
Solving x² − 2x − 3 = 0. The curve y = x² − 2x − 3 has the rule on one side of the equation. The equation says that this rule equals 0. On the graph, y = 0 is the x-axis. So the solutions are the x-values where the curve crosses the x-axis. They are the roots.
Solving x² − 2x − 3 = 5. Now the rule must equal 5. Draw the horizontal line y = 5 across the grid. Where it meets the curve, the curve has y = 5. Read the x-values of those points. There can be two, one or no solutions.
Solving x² = 2x + 3. The two sides are two different rules. Draw both as graphs on the same grid: the curve y = x² and the straight line y = 2x + 3. At a crossing point both graphs have the same x and the same y, so x² = 2x + 3 is true there. The solutions are the x-values of the crossing points.
Answers read from a graph are only approximate: you can only read as accurately as the grid allows. If the crossing is between grid lines, estimate to 1 decimal place. Exam questions that need this will say “use the graph” or “give your answers to 1 decimal place”. Always show the line you drew.
Key ideas
- A solution of an equation is the x-value of a point where two graphs meet.
- To solve (rule) = 0, read where the curve crosses the x-axis.
- To solve (rule) = k, draw the horizontal line y = k and read the x-values where it meets the curve.
- To solve (rule 1) = (rule 2), graph both on the same grid and read the x-values where they cross.
- Graph answers are approximate. Read the x-values, not the y-values.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2 − 2x − 3. The equation we want to solve is the same expression set equal to 0.x2 − 2x − 3 = 0 means y = 0 on this graph
- Draw the horizontal line y = 0. Every point on this line has a y-value of 0. Where the line meets the curve, the curve has y = 0, which is exactly when the equation is true.Start at 0 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−1, 0) and (3, 0).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = −1 or 3
Check. Substitute back: y = (−1)2 − 2 × (−1) − 3 = 1 + 2 − 3 = 0; y = 32 − 2 × 3 − 3 = 9 − 6 − 3 = 0 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2 − 2x − 3. The equation we want to solve is the same expression set equal to 5.x2 − 2x − 3 = 5 means y = 5 on this graph
- Draw the horizontal line y = 5. Every point on this line has a y-value of 5. Where the line meets the curve, the curve has y = 5, which is exactly when the equation is true.Start at 5 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−2, 5) and (4, 5).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = −2 or 4
Check. Substitute back: y = (−2)2 − 2 × (−2) − 3 = 4 + 4 − 3 = 5; y = 42 − 2 × 4 − 3 = 16 − 8 − 3 = 5 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. The solutions are the x-values of the points where the two graphs cross.
- Say which two graphs are being used. The equation has the rule of the curve on one side and the rule of the line on the other. Each side is drawn as its own graph.Curve: y = x2
Line: y = 2x + 3 - Explain why crossing points give the answers. At a crossing point both graphs have the same x and the same y, so the two rules give the same value. That is what the equation says.Solve x² = 2x + 3: the curve y = x² and the line y = 2x + 3 are both drawn.
- Find the crossing points. Look for every place where the two graphs meet.They cross at (−1, 1) and (3, 9).
- Read the x-values. The solutions are the x-coordinates of the crossing points.x = −1 or 3
Check. Substitute each x-value into both sides.
x = −1: left = 1, right = 1. Equal. ✓
x = 3: left = 9, right = 9. Equal. ✓
Watch out. Write only the x-values. (The y-values at the crossings are not the answer.)
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2. The equation we want to solve is the same expression set equal to 10.x2 = 10 means y = 10 on this graph
- Draw the horizontal line y = 10. Every point on this line has a y-value of 10. Where the line meets the curve, the curve has y = 10, which is exactly when the equation is true.Start at 10 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−3.2, 10) and (3.2, 10).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x ≈ −3.2 or 3.2 (accept answers within 0.1)
Check. Substitute back: at x = −3.2 the rule gives 10.00; at x = 3.2 the rule gives 10.00 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Watch out
Read the x-values where the graphs meet. The y-value of a crossing point is not the answer. For x² = 2x + 3 the crossing points are (−1, 1) and (3, 9), but the solutions are x = −1 and x = 3. Also remember there can be two solutions: look for both crossings.Exam tip
Draw the line you use in a different colour with a ruler, and put a small cross at each crossing point before you read down to the x-axis. Write “x = ... or x = ...”.Exercise 13C Solving equations using graphs
Non-calculatorKey words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2 − 2x − 3. The equation we want to solve is the same expression set equal to 0.x2 − 2x − 3 = 0 means y = 0 on this graph
- Draw the horizontal line y = 0. Every point on this line has a y-value of 0. Where the line meets the curve, the curve has y = 0, which is exactly when the equation is true.Start at 0 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−1, 0) and (3, 0).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = −1 or 3
Check. Substitute back: y = (−1)2 − 2 × (−1) − 3 = 1 + 2 − 3 = 0; y = 32 − 2 × 3 − 3 = 9 − 6 − 3 = 0 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2 − 2x − 3. The equation we want to solve is the same expression set equal to 5.x2 − 2x − 3 = 5 means y = 5 on this graph
- Draw the horizontal line y = 5. Every point on this line has a y-value of 5. Where the line meets the curve, the curve has y = 5, which is exactly when the equation is true.Start at 5 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−2, 5) and (4, 5).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = −2 or 4
Check. Substitute back: y = (−2)2 − 2 × (−2) − 3 = 4 + 4 − 3 = 5; y = 42 − 2 × 4 − 3 = 16 − 8 − 3 = 5 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2 − 2x − 3. The equation we want to solve is the same expression set equal to −3.x2 − 2x − 3 = −3 means y = −3 on this graph
- Draw the horizontal line y = −3. Every point on this line has a y-value of −3. Where the line meets the curve, the curve has y = −3, which is exactly when the equation is true.Start at −3 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (0, −3) and (2, −3).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = 0 or 2
Check. Substitute back: y = 02 − 2 × 0 − 3 = 0 + 0 − 3 = −3; y = 22 − 2 × 2 − 3 = 4 − 4 − 3 = −3 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2 − 2x − 3. The equation we want to solve is the same expression set equal to −4.x2 − 2x − 3 = −4 means y = −4 on this graph
- Draw the horizontal line y = −4. Every point on this line has a y-value of −4. Where the line meets the curve, the curve has y = −4, which is exactly when the equation is true.Start at −4 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (1, −4).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = 1
Check. Substitute back: y = 12 − 2 × 1 − 3 = 1 − 2 − 3 = −4 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2. The equation we want to solve is the same expression set equal to 4.x2 = 4 means y = 4 on this graph
- Draw the horizontal line y = 4. Every point on this line has a y-value of 4. Where the line meets the curve, the curve has y = 4, which is exactly when the equation is true.Start at 4 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−2, 4) and (2, 4).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = −2 or 2
Check. Substitute back: y = (−2)2 = 4; y = 22 = 4 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2. The equation we want to solve is the same expression set equal to 9.x2 = 9 means y = 9 on this graph
- Draw the horizontal line y = 9. Every point on this line has a y-value of 9. Where the line meets the curve, the curve has y = 9, which is exactly when the equation is true.Start at 9 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−3, 9) and (3, 9).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = −3 or 3
Check. Substitute back: y = (−3)2 = 9; y = 32 = 9 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2. The equation we want to solve is the same expression set equal to 6.25.x2 = 6.25 means y = 6.25 on this graph
- Draw the horizontal line y = 6.25. Every point on this line has a y-value of 6.25. Where the line meets the curve, the curve has y = 6.25, which is exactly when the equation is true.Start at 6.25 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−2.5, 6.25) and (2.5, 6.25).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = −2.5 or 2.5
Check. Substitute back: y = (−2.5)2 = 6.25; y = 2.52 = 6.25 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2. The equation we want to solve is the same expression set equal to 10.x2 = 10 means y = 10 on this graph
- Draw the horizontal line y = 10. Every point on this line has a y-value of 10. Where the line meets the curve, the curve has y = 10, which is exactly when the equation is true.Start at 10 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−3.2, 10) and (3.2, 10).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x ≈ −3.2 or 3.2 (accept answers within 0.1)
Check. Substitute back: at x = −3.2 the rule gives 10.00; at x = 3.2 the rule gives 10.00 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2 − 4x + 3. The equation we want to solve is the same expression set equal to 0.x2 − 4x + 3 = 0 means y = 0 on this graph
- Draw the horizontal line y = 0. Every point on this line has a y-value of 0. Where the line meets the curve, the curve has y = 0, which is exactly when the equation is true.Start at 0 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (1, 0) and (3, 0).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = 1 or 3
Check. Substitute back: y = 12 − 4 × 1 + 3 = 1 − 4 + 3 = 0; y = 32 − 4 × 3 + 3 = 9 − 12 + 3 = 0 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2 − 4x + 3. The equation we want to solve is the same expression set equal to 3.x2 − 4x + 3 = 3 means y = 3 on this graph
- Draw the horizontal line y = 3. Every point on this line has a y-value of 3. Where the line meets the curve, the curve has y = 3, which is exactly when the equation is true.Start at 3 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (0, 3) and (4, 3).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = 0 or 4
Check. Substitute back: y = 02 − 4 × 0 + 3 = 0 + 0 + 3 = 3; y = 42 − 4 × 4 + 3 = 16 − 16 + 3 = 3 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2 − 4x + 3. The equation we want to solve is the same expression set equal to 8.x2 − 4x + 3 = 8 means y = 8 on this graph
- Draw the horizontal line y = 8. Every point on this line has a y-value of 8. Where the line meets the curve, the curve has y = 8, which is exactly when the equation is true.Start at 8 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−1, 8) and (5, 8).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = −1 or 5
Check. Substitute back: y = (−1)2 − 4 × (−1) + 3 = 1 + 4 + 3 = 8; y = 52 − 4 × 5 + 3 = 25 − 20 + 3 = 8 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2 − 4x + 3. The equation we want to solve is the same expression set equal to −1.x2 − 4x + 3 = −1 means y = −1 on this graph
- Draw the horizontal line y = −1. Every point on this line has a y-value of −1. Where the line meets the curve, the curve has y = −1, which is exactly when the equation is true.Start at −1 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (2, −1).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = 2
Check. Substitute back: y = 22 − 4 × 2 + 3 = 4 − 8 + 3 = −1 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = −x2 + 4. The equation we want to solve is the same expression set equal to 0.4 − x² = 0 means y = 0 on this graph
- Draw the horizontal line y = 0. Every point on this line has a y-value of 0. Where the line meets the curve, the curve has y = 0, which is exactly when the equation is true.Start at 0 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−2, 0) and (2, 0).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = −2 or 2
Check. Substitute back: y = −(−2)2 + 4 = −4 + 4 = 0; y = −(2)2 + 4 = −4 + 4 = 0 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = −x2 + 4. The equation we want to solve is the same expression set equal to 3.4 − x² = 3 means y = 3 on this graph
- Draw the horizontal line y = 3. Every point on this line has a y-value of 3. Where the line meets the curve, the curve has y = 3, which is exactly when the equation is true.Start at 3 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−1, 3) and (1, 3).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = −1 or 1
Check. Substitute back: y = −(−1)2 + 4 = −1 + 4 = 3; y = −(1)2 + 4 = −1 + 4 = 3 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = −x2 + 4. The equation we want to solve is the same expression set equal to −5.4 − x² = −5 means y = −5 on this graph
- Draw the horizontal line y = −5. Every point on this line has a y-value of −5. Where the line meets the curve, the curve has y = −5, which is exactly when the equation is true.Start at −5 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−3, −5) and (3, −5).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = −3 or 3
Check. Substitute back: y = −(−3)2 + 4 = −9 + 4 = −5; y = −(3)2 + 4 = −9 + 4 = −5 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = −x2 + 4. The equation we want to solve is the same expression set equal to 4.4 − x² = 4 means y = 4 on this graph
- Draw the horizontal line y = 4. Every point on this line has a y-value of 4. Where the line meets the curve, the curve has y = 4, which is exactly when the equation is true.Start at 4 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (0, 4).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x = 0
Check. Substitute back: y = −(0)2 + 4 = 0 + 4 = 4 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. The solutions are the x-values of the points where the two graphs cross.
- Say which two graphs are being used. The equation has the rule of the curve on one side and the rule of the line on the other. Each side is drawn as its own graph.Curve: y = x2
Line: y = 2x + 3 - Explain why crossing points give the answers. At a crossing point both graphs have the same x and the same y, so the two rules give the same value. That is what the equation says.Solve x² = 2x + 3.
- Find the crossing points. Look for every place where the two graphs meet.They cross at (−1, 1) and (3, 9).
- Read the x-values. The solutions are the x-coordinates of the crossing points.x = −1 or 3
Check. Substitute each x-value into both sides.
x = −1: left = 1, right = 1. Equal. ✓
x = 3: left = 9, right = 9. Equal. ✓
Watch out. Write only the x-values. (The y-values at the crossings are not the answer.)
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. The solutions are the x-values of the points where the two graphs cross.
- Say which two graphs are being used. The equation has the rule of the curve on one side and the rule of the line on the other. Each side is drawn as its own graph.Curve: y = x2
Line: y = x + 2 - Explain why crossing points give the answers. At a crossing point both graphs have the same x and the same y, so the two rules give the same value. That is what the equation says.Solve x² = x + 2.
- Find the crossing points. Look for every place where the two graphs meet.They cross at (−1, 1) and (2, 4).
- Read the x-values. The solutions are the x-coordinates of the crossing points.x = −1 or 2
Check. Substitute each x-value into both sides.
x = −1: left = 1, right = 1. Equal. ✓
x = 2: left = 4, right = 4. Equal. ✓
Watch out. Write only the x-values. (The y-values at the crossings are not the answer.)
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. The solutions are the x-values of the points where the two graphs cross.
- Say which two graphs are being used. The equation has the rule of the curve on one side and the rule of the line on the other. Each side is drawn as its own graph.Curve: y = x2 − 4
Line: y = x − 2 - Explain why crossing points give the answers. At a crossing point both graphs have the same x and the same y, so the two rules give the same value. That is what the equation says.Solve x² − 4 = x − 2.
- Find the crossing points. Look for every place where the two graphs meet.They cross at (−1, −3) and (2, 0).
- Read the x-values. The solutions are the x-coordinates of the crossing points.x = −1 or 2
Check. Substitute each x-value into both sides.
x = −1: left = −3, right = −3. Equal. ✓
x = 2: left = 0, right = 0. Equal. ✓
Watch out. Write only the x-values. (The y-values at the crossings are not the answer.)
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2 − 2x − 3. The equation we want to solve is the same expression set equal to 7.x2 − 2x − 3 = 7 means y = 7 on this graph
- Draw the horizontal line y = 7. Every point on this line has a y-value of 7. Where the line meets the curve, the curve has y = 7, which is exactly when the equation is true.Start at 7 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−2.3, 7) and (4.3, 7).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x ≈ −2.3 or 4.3 (accept answers within 0.1)
Check. Substitute back: at x = −2.3 the rule gives 7.00; at x = 4.3 the rule gives 7.00 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the horizontal line y = (the number on the right). Where it crosses the curve, read the x-values.
- Spot the graph being used. The graph shows y = x2 − 3x. The equation we want to solve is the same expression set equal to 1.x2 − 3x = 1 means y = 1 on this graph
- Draw the horizontal line y = 1. Every point on this line has a y-value of 1. Where the line meets the curve, the curve has y = 1, which is exactly when the equation is true.Start at 1 on the y-axis and draw straight across (a ruler helps).
- Find where the line crosses the curve. Mark each crossing point.The line meets the curve at (−0.3, 1) and (3.3, 1).
- Read the x-values. The solutions of the equation are the x-values of the crossing points (the numbers along the bottom axis).x ≈ −0.3 or 3.3 (accept answers within 0.1)
Check. Substitute back: at x = −0.3 the rule gives 1.00; at x = 3.3 the rule gives 1.00 ✓
Watch out. Read the x-values of the crossing points, not the y-values. The y-value is just the number you started with.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Connect "the rule equals 0" with "y equals 0 on the graph".
- Write the graph's equation. The curve is y = x² − 2x − 3.y = x² − 2x − 3
- Write the equation to be solved. It says the same rule equals 0.x² − 2x − 3 = 0
- Match them. If the rule equals 0 then y = 0. Points with y = 0 are on the x-axis.So the solutions are the x-values where the curve meets the x-axis, which are called the roots.
Check. The graph meets the x-axis at x = −1 and x = 3, and substituting them gives 0. ✓
Watch out. Roots, x-intercepts and "solutions of = 0" are all the same thing.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. A solution is an x-value, so read the x-coordinates of the crossing points.
- Say what Emma has read. 1 and 9 are the y-coordinates of the crossing points.Emma read the y-values.
- Say what is needed. The equation is in x, so each solution is a value of x that makes both sides equal.We need the x-coordinates.
- Read the x-coordinates. The points are (−1, 1) and (3, 9).x = −1 and x = 3
Check. x = −1: left = (−1)² = 1, right = 2 × (−1) + 3 = 1. Equal. ✓
x = 3: left = 9, right = 2 × 3 + 3 = 9. Equal. ✓
Watch out. Solutions are x-values. y-values are not solutions.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw the line y = −5 and see whether it meets the curve.
- Draw the line. The line y = −5 is horizontal, 5 units below the x-axis.y = −5
- Look at the curve. The curve y = x² never goes below the x-axis (squares are never negative). Its lowest point is (0, 0).
- Conclude. The line and curve never meet, so there is no x-value for which x² = −5.No solutions.
Check. Squaring any number gives 0 or a positive number, never −5. ✓
Watch out. "No solutions" is a valid answer.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw y = 2 and see how often it meets the curve.
- Compare k with the lowest value. The lowest point of the curve is y = −4. The line y = 2 is above it.2 > −4
- Count crossings. A horizontal line above the lowest point of a U-shaped curve crosses it twice.2 solutions
Check. x² − 2x − 5 = 0 has two real solutions (1 ± √6). ✓
Watch out. Compare k with the turning point.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw y = −4 and see how often it meets the curve.
- Compare k with the lowest value. The lowest point is exactly y = −4.−4 = −4
- Count crossings. The line just touches the curve at the lowest point.1 solution (x = 1)
Check. x² − 2x + 1 = (x − 1)² = 0 gives only x = 1. ✓
Watch out. Touching counts as one solution.
Key words. To solve an equation graphically means to read its answers from a graph. Where two graphs meet they have the same x and the same y, so the x-value at a crossing point makes the two rules equal. Graph readings are approximate because we can only read to the accuracy of the drawing.
Method. Draw y = −6 and see whether it meets the curve.
- Compare k with the lowest value. The line y = −6 is below the lowest point y = −4.−6 < −4
- Count crossings. The curve never goes below −4, so the line never meets it.No solutions
Check. The lowest value of the rule is −4, so it can never equal −6. ✓
Watch out. Below the lowest point means no crossing.