📝 Solutions of equation and graph relationship (13 MCQs)

📖 From Digital SAT Algebra • 4. Graphs • 13 questions available

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What is Solutions of equation and graph relationship?

Definition:
The relationship between solutions of an equation and its graph is that every ordered pair (x, y) that satisfies the equation corresponds to a point on the graph of the equation, and conversely, every point on the graph represents a solution to the equation; thus, the graph is a visual representation of the solution set, and for linear equations, this solution set forms a straight line, providing a geometric interpretation of algebraic solutions.

Working:
When we graph a linear equation, we are plotting all points that satisfy the equation; the x and y coordinates of each point on the line make the equation true; for example, for $y = 2x$ , the point (1, 2) satisfies $2 = 2(1)$ , and (2, 4) satisfies $4 = 2(2)$ ; thus, the line is the set of all solutions; this connection is used to solve equations graphically, where the intersection of two graphs gives the solution to a system, and it helps interpret real-world situations.

Example:
A simple example is the equation $y = x + 1$ ; its graph is a line passing through (0, 1), (1, 2), (2, 3); each of these points is a solution, and any point on the line, such as (3, 4) or (-1, 0), is also a solution; this shows the direct correspondence between solutions and the graph, illustrating the relationship.

Reason:
Understanding the relationship between equations and graphs is central to algebra, as it bridges algebraic and geometric thinking, enables solving systems, and helps model and analyze real-world problems, making it a foundational concept in mathematics.

Easy

Medium

Hard

📝 All Solutions of equation and graph relationship MCQs

Q1. A student solves $2x+6=0$ and obtains $x=-3$ . Which statement best connects this algebraic solution to the graph of $y=2x+6$ ?

A.The graph crosses the

x

-axis at

(-3,0)

, because the solution makes

y=0

. ✅

B.The graph crosses the

y

-axis at

(-3,0)

, because

x

is the solution.

C.The graph has a slope of

-3

, because the solution is negative.

D.The graph never reaches the

x

-axis because the solution is negative.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The equation $2x+6=0$ gives $x=-3$ . On the graph of $y=2x+6$ , setting $y=0$ identifies an $x$ -intercept. Therefore, the algebraic solution corresponds directly to the point $(-3,0)$ where the graph crosses the $x$ -axis.

Q2. For an equation written as $f(x)=0$ , what does each real solution represent on the graph of $y=f(x)$ ?

A.An

x

-coordinate where the graph intersects the

x

-axis ✅

B.A

y

-coordinate where the graph intersects the

y

-axis

C.The slope of the graph at the solution

D.The highest

y

-value of the graph

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: When $f(x)=0$ , the corresponding point on $y=f(x)$ has coordinates $(x,0)$ . Thus, every real solution is an $x$ -coordinate of an $x$ -axis intersection. This interpretation connects solving an equation with identifying graph intersections.

Q3. A graph of $y=x^2-5x+6$ crosses the $x$ -axis at $x=2$ and $x=3$ . Without factoring the expression, which conclusion is justified?

A.The equation

x^2-5x+6=0

has exactly two real solutions,

2

and

3

. ✅

B.The equation has only one real solution because the graph is a parabola.

C.The solutions must be

x=-2

and

x=-3

because the graph opens upward.

D.The equation has no solutions because the graph does not cross the

y

-axis at zero.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: An $x$ -axis intersection occurs where $y=0$ . Since the graph crosses the axis at $x=2$ and $x=3$ , those values satisfy $x^2-5x+6=0$ . Two distinct intersections therefore indicate exactly two real solutions.

Q4. Two students solve $x^2-4x+4=0$ . Ali says there are two solutions because the equation has degree two. Sara says there is one solution because the graph touches the $x$ -axis once. Who is correct?

A.Ali, because every quadratic equation must have two distinct real solutions.

B.Sara, because one

x

-axis contact corresponds to one real solution. ✅

C.Both, because a touching point represents two different

x

-values.

D.Neither, because solutions cannot be identified graphically.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A quadratic can have one repeated real solution. Here the graph touches the $x$ -axis at a single $x$ -value rather than crossing it. That single $x$ -coordinate represents one distinct real solution, even though its multiplicity may be two.

Q5. A company models profit with $P(x)=x^2-8x+15$ , where $x$ represents a production level. At which production levels is the company breaking even?

A.At

x=3

and

x=5

✅

B.At

x=0

and

x=15

C.At

x=4

only

D.At

x=8

only

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Breaking even means profit equals zero, so solve $P(x)=0$ . The expression factors as $(x-3)(x-5)=0$ , giving $x=3$ and $x=5$ . Graphically, these are exactly the two points where the profit graph meets the $x$ -axis.

Q6. The equation $3x-9=0$ is represented by $y=3x-9$ . A student claims that $x=3$ is the $y$ -intercept because it is the solution. What is the best correction?

A.The solution

x=3

gives the

x

-intercept

(3,0)

, while the

y

-intercept occurs when

x=0

. ✅

B.The solution

x=3

gives the

y

-intercept

(0,3)

C.The solution must always represent the slope.

D.There is no relationship between a solution and an intercept.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: A solution to $3x-9=0$ makes the function value zero, so it identifies an $x$ -intercept. Substituting $x=3$ gives $y=0$ , producing $(3,0)$ . The $y$ -intercept instead requires $x=0$ , giving $(0,-9)$ .

Q7. A temperature model is $T(t)=2t-10$ , where $t$ is time in hours. The graph crosses the horizontal axis at $t=5$ . What does this intersection mean in the model?

A.The temperature is

0

t=5

hours. ✅

B.The temperature reaches

5^\circ

t=0

C.The temperature increases by

5^\circ

every hour.

D.The initial temperature is

5^\circ

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The horizontal axis represents $T=0$ . Therefore, the intersection at $t=5$ means $T(5)=0$ . In the context of the model, the temperature reaches $0^\circ$ five hours after the starting time.

Q8. A graph of $y=f(x)$ stays entirely above the $x$ -axis. A student concludes that $f(x)=0$ has a solution because every function must equal zero somewhere. How should the reasoning be evaluated?

A.The reasoning is correct because every graph eventually crosses the

x

-axis.

B.The reasoning is incorrect because the graph has no point with

y=0

. ✅

C.The reasoning is correct if the graph has a positive slope.

D.The reasoning is incorrect only when the graph is nonlinear.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Solutions of $f(x)=0$ correspond to points where the graph has $y=0$ . If the entire graph remains above the $x$ -axis, no such point exists. Therefore, the equation has no real solution represented by the graph.

Q9. A student estimates that a graph crosses the $x$ -axis near $x=1.8$ and $x=4.2$ . Another student solves the corresponding equation algebraically and obtains $x=1.79$ and $x=4.21$ . Which conclusion is most reasonable?

A.The methods disagree because graphical estimates can never represent solutions.

B.The graphical and algebraic methods are consistent because the graph gives approximate values. ✅

C.The algebraic answers must be wrong because they are not whole numbers.

D.Only the graph can determine whether the solutions are real.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Graphs often provide approximate coordinates because values are read from a visual scale, while algebraic methods can provide greater precision. Values near $1.8$ and $4.2$ agree closely with $1.79$ and $4.21$ , so the methods support the same solutions.

Q10. A graph intersects the $x$ -axis at $x=-4$ , touches it at $x=1$ , and intersects it again at $x=6$ . How many distinct real solutions does the equation represented by $y=f(x)$ have?

A.One

B.Two

C.Three ✅

D.Six

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Every distinct $x$ -coordinate where the graph has $y=0$ represents one distinct real solution. The graph has three such $x$ -values: $-4$ , $1$ , and $6$ . Whether the graph crosses or merely touches the axis does not change the number of distinct solutions.

Q11. A student says, \"If a graph crosses the $y$ -axis at $4$ then the equation has the solution $x=4$ \". Which response best identifies the error?

A.The student confused the

y

-intercept with an

x

-intercept; a solution of

f(x)=0

is identified by an

x

-axis intersection. ✅

B.The student should have used the slope instead of the

y

-intercept.

C.The student is correct because every intercept is a solution.

D.The student should replace

x=4

with

y=4

as the only possible solution.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: A $y$ -axis intersection occurs when $x=0$ , so a point such as $(0,4)$ does not mean $x=4$ is a solution. For $f(x)=0$ , the relevant graph feature is an $x$ -axis intersection, where the output equals zero.

Q12. Suppose $g(x)=0$ has no real solutions. What must be true about the graph of $y=g(x)$ , assuming the graph represents all real $x$ -values?

A.It never intersects the

x

-axis. ✅

B.It must cross the

y

-axis exactly twice.

C.It must have zero slope everywhere.

D.It must contain the point

(0,0)

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: A real solution of $g(x)=0$ would create a point $(x,0)$ on the graph. If there are no real solutions, the graph cannot contain any point on the $x$ -axis. It may still intersect the $y$ -axis elsewhere.

Q13. Consider $h(x)=(x-2)^2-9$ . A graphing approach shows two $x$ -axis intersections. Which pair of values should be expected, and why?

x=-1

and

x=5

, because setting

h(x)=0

gives

(x-2)^2=9

. ✅

x=2

and

x=9

, because these numbers appear in the expression.

x=-2

and

x=2

, because the squared term is centered at

2

x=3

and

x=11

, because the constant is added to

2

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: To find the $x$ -axis intersections, set $h(x)=0$ : $(x-2)^2-9=0$ , so $(x-2)^2=9$ . Thus $x-2=\pm3$ , producing $x=-1$ and $x=5$ . These are the two real solutions represented by the graph.

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