Exercises: Understand Graphs of Equations

Show all substitution work when verifying points. For graphing problems, generate a table of values before plotting.

Grade 9·23 problems·~30 min·Common Core Math - HS Algebra·standard·hsa-rei-d-10

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Warm-Up: Review What You Know

These problems review skills you already have.

Evaluate $y = 3x - 1$ when $x = 4$ .

Which statement correctly describes the solution to the one-variable equation $x = 5$ ?

The solution is the single point $x = 5$ on a number line.

The solution is the line $y = 5$ in the coordinate plane.

The solution is every ordered pair $(5, y)$ for any value of $y$ .

The solution is every ordered pair $(x, 5)$ for any value of $x$ .

A student says the ordered pair $(2, 7)$ satisfies the equation $y = 3x + 1$ .
Is the student correct?

Yes, because $3(2) + 1 = 7$ .

No, because $2 + 7 \neq 1$ .

Yes, because $2 + 7 = 9$ and $3 + 1 = 4$ .

No, because $3(7) + 1 \neq 2$ .

Determine whether each given point lies on the graph of the equation. Show your substitution.

Does the point $(3, 10)$ lie on the graph of $y = 2x + 4$ ?

Yes — substitute $x = 3$ : $2(3) + 4 = 10$ ✓

No — the graph only passes through the origin.

Yes — $3 + 10 = 13$ and $2 + 4 = 6$ , so they are close.

No — $3 \times 10 = 30$ , which is not equal to $2(3) + 4$ .

Which of the following points is NOT on the graph of $y = x^2 - 1$ ?

$(0, -1)$

$(2, 3)$

$(3, 8)$

$(-2, 5)$

Is the point $(3, 4)$ on the graph of $x^2 + y^2 = 25$ ?

Yes — $3^2 + 4^2 = 9 + 16 = 25$ ✓

No — circles don't pass through points with equal $x$ and $y$ values.

Yes — $3 + 4 = 7$ and $25 \div 7 \approx 3.6$ , which is close.

No — $3^2 + 4^2 = 12 + 16 = 28 \neq 25$ .

The point $(-3, y)$ lies on the graph of $y = x^2 + 2x - 5$ . What is the value of $y$ ?

The graph of $y = 2^x$ is an exponential curve. Which of the following points lies on this graph?

$(0, 0)$

$(3, 6)$

$(4, 16)$

$(-1, -2)$

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