Open Practice Tests

Graphing equations and finding intersections — Topic practice

5 exercises · 3 automatically checked · 2 written self-checks · Untimed

Instructions and review guidance

3 questions · Shuffled order · Untimed practice

Try each question before choosing Check answer. Guide question labels stay the same when you shuffle the bank. 2 written responses appear below the bank, with model answers for self-review; these are not automatically graded. Follow any rounding instruction in the question.

Guide question labels stay the same when the order changes. Check answers when you are ready to review.

Written self-checks are below the question bank and are excluded from the automatic score.

Written responses are self-review exercises.

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Compare the reasoning as well as the answer. Equivalent valid methods are welcome. A referenced example provides a targeted repair route.

The drill

Question 1

  • Guide question E2.1

The equations x + y = 9 and x − y = 3 have solution (x, y). Find y.

Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.

Question 3

  • Guide question E2.3

The circle (x + 1)2 + (y − 2)2 = 25 intersects the line y = 5. Find the greater x-coordinate of an intersection.

Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.

Question 5

  • Guide question E2.5

For which value of k does y = k intersect y = x2 − 8x + 21 at exactly one point?

Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.

Question navigator and options
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Written self-checks

Write your complete response before opening the model answer. Compare both your answer and your reasoning.

Guide question E2.2

What is the greater solution of x2 − 7x + 10 = 0? Describe which graph feature represents it.

Show model answerHide model answer for question E2.2

5.

The graph y = x2 − 7x + 10 has x-intercepts 2 and 5. The greater input is 5. Factoring gives (x − 2)(x − 5) = 0, confirming both roots and their ordering. The vertex is not the solution feature requested here; a root is an input where the output equals zero. Review Example 2.03.

Guide question E2.4

Solve x + 3 = x − 3 over the real numbers. Explain why a squared equation alone is insufficient.

Show model answerHide model answer for question E2.4

6.

The original right side must be nonnegative, so x ≥ 3. Squaring gives x +3 = x2 −6x +9, or (x −1)(x −6) = 0. The candidate 1 fails the original equation because 2 ≠ −2; the candidate 6 works because 3 = 3. The only solution is 6. Review Example 2.11.