Unit 10 Class 9 Math Solutions – Graphs of Functions
Unit 10 Class 9 Math Solutions contain complete exercise-wise PDF solutions of the chapter Graphs of Functions. Students can find the solutions of Exercise 10.1, Exercise 10.2, and Review Exercise 10 on this page.
This unit explains how to sketch and interpret linear, quadratic, cubic, exponential, reciprocal, square-root, and cube-root graphs. It also covers tangents, gradients, domains, ranges, asymptotes, intercepts, vertices, and practical applications of graphs.
The solution PDFs include value tables, carefully drawn graphs, equations, graph features, interpretations, application questions, and complete step-by-step calculations for the Punjab Board Class 9 Mathematics book.
Students can view the PDFs online and use the save or download option in the PDF viewer to keep the solutions for offline study.
These Unit 10 solutions are useful for homework, revision, class tests, annual examinations, and board exam preparation.
Solutions of Exercise 10.1 Unit 10 Class 9 Math Notes
Exercise 10.1 of Unit 10 Class 9 Math Notes focuses on drawing and understanding different types of function graphs. Students prepare value tables, find intercepts, identify important graph features, and sketch each curve accurately.
How to Draw a Graph from an Equation
The usual method is to choose suitable values of the independent variable, calculate the corresponding function values, plot the ordered pairs, and join them according to the type of graph.
Straight-line points are joined with a ruler.
Polynomial, exponential, reciprocal, and root-function points are joined with smooth curves.
Table of Values
A table of values helps students organize the coordinates before plotting.
For a function written as \(y=f(x)\), select values of \(x\) and calculate:
\(y=f(x)\)
The chosen interval should include important points such as intercepts, vertices, and values on both sides of turning points or asymptotes.
Straight-Line Graphs
A linear function has the form:
\(y=mx+c\)
Here, \(m\) is the slope or gradient, and \(c\) is the vertical intercept.
The graph is one straight line.
Slope of a Straight Line
A positive slope means that the line rises from left to right.
A negative slope means that the line falls from left to right.
A larger absolute value of the slope produces a steeper line.
Vertical Intercept
The vertical intercept is found by putting \(x=0\).
For:
\(y=mx+c\)
the graph crosses the vertical axis at:
\((0,c)\)
Horizontal Intercept
The horizontal intercept is found by putting \(y=0\) and solving for \(x\).
For example:
\(3x-5=0\)
gives:
\(x=\frac{5}{3}\)
Graph of a Quadratic Function
A quadratic function has the form:
\(y=ax^2+bx+c\)
Its graph is a parabola.
The parabola opens upward when \(a>0\) and downward when \(a<0\).
Axis of Symmetry
The axis of symmetry of a quadratic graph is:
\(x=-\frac{b}{2a}\)
The vertex lies on this vertical line.
Vertex of a Parabola
First calculate the axis of symmetry.
Then substitute its value into the function to find the corresponding vertical coordinate.
The resulting point is the minimum point of an upward-opening parabola or the maximum point of a downward-opening parabola.
Roots of a Quadratic Graph
The roots or horizontal intercepts are the values of \(x\) for which:
\(y=0\)
Factoring is often the quickest method when the quadratic expression can be written as a product of two linear factors.
Cubic Graphs
A cubic function has highest power 3.
Its graph may cross the horizontal axis up to three times.
Factoring helps identify the roots before a value table is prepared.
The points should be joined with one smooth cubic curve.
Factored Form of a Cubic
If:
\(y=(x-a)(x-b)(x-c)\)
then the horizontal intercepts are:
\((a,0),\qquad(b,0),\qquad(c,0)\)
The vertical intercept is found by putting \(x=0\).
Leading Coefficient and End Behaviour
A cubic graph with a positive leading coefficient generally falls toward the left and rises toward the right.
A cubic graph with a negative leading coefficient generally rises toward the left and falls toward the right.
Exponential Growth
An exponential function with base greater than 1 has the form:
\(y=a^x,\qquad a>1\)
Its graph increases as \(x\) increases.
The graph passes through:
\((0,1)\)
because:
\(a^0=1\)
Exponential Decay
An exponential decay function may be written as:
\(y=a^{-x}=\left(\frac{1}{a}\right)^x,\qquad a>1\)
Its graph decreases as \(x\) increases but remains above the horizontal axis.
Domain and Range of Exponential Functions
For ordinary positive-base exponential functions, the domain is all real numbers.
The range is:
\(y>0\)
The horizontal asymptote is:
\(y=0\)
Reciprocal Functions
A reciprocal function places the variable in the denominator.
A basic example is:
\(y=\frac{1}{x}\)
The graph usually has two separate branches.
Vertical Asymptote of a Reciprocal Function
A vertical asymptote occurs where the denominator is zero.
For:
\(y=\frac{1}{x-3}\)
the vertical asymptote is:
\(x=3\)
The function is not defined at this value.
Horizontal Asymptote of a Reciprocal Function
For a function such as:
\(y=\frac{1}{x-3}\)
the function approaches zero when the absolute value of \(x\) becomes large.
Therefore, the horizontal asymptote is:
\(y=0\)
Shifted Reciprocal Graphs
A reciprocal graph can be shifted vertically.
For:
\(y=\frac{2}{x}+3\)
the vertical asymptote is \(x=0\), while the horizontal asymptote is:
\(y=3\)
Its range excludes the horizontal-asymptote value.
Domain and Range of Reciprocal Functions
Values that make the denominator zero must be excluded from the domain.
The horizontal-asymptote value is normally excluded from the range.
For:
\(y=\frac{1}{x-3}\)
the domain and range are:
\(x\neq3,\qquad y\neq0\)
Square-Root Graph
The square-root function is:
\(y=\sqrt{x}=x^{1/2}\)
It is defined only for:
\(x\geq0\)
Its range is:
\(y\geq0\)
The graph starts at the origin and increases gradually.
Cube-Root Graph
A cube-root function is defined for positive, zero, and negative values.
The exercise includes:
\(y=3\sqrt[3]{x}\)
Its domain and range are all real numbers.
It passes through the origin and is symmetric about the origin.
Inverse-Square Graph
Exercise 10.1 also includes:
\(y=2x^{-2}=\frac{2}{x^2}\)
The function is undefined at \(x=0\).
Both branches lie above the horizontal axis because \(x^2>0\) for \(x eq0\).
Its asymptotes are:
\(x=0\)
and:
\(y=0\)
The graph is symmetric about the vertical axis.
Graph Symmetry
A graph is symmetric about the vertical axis when replacing \(x\) by \(-x\) does not change the function value.
A graph is symmetric about the origin when:
\(f(-x)=-f(x)\)
Recognizing symmetry reduces the number of points that must be calculated.
Solutions of Exercise 10.2 Unit 10 Class 9 Math Notes
Exercise 10.2 applies graphs to tangents, changing quantities, demand and supply, salaries, costs, revenues, break-even points, profits, and other real-life situations.
Gradient of a Tangent
The gradient of the tangent to a curve at one point represents the instantaneous rate of change.
For \(y=f(x)\), the tangent gradient at \(x=a\) is:
\(m=f'(a)\)
Equation of a Tangent
A tangent through the point \((a,f(a))\) with gradient \(m\) is written in point-slope form:
\(y-f(a)=m(x-a)\)
After substitution, simplify the equation into straight-line form.
Tangent to a Quadratic Curve
Differentiate the quadratic function to obtain its gradient function.
Substitute the horizontal coordinate of the point into the derivative.
Then use the point and gradient in the tangent equation.
Checking a Tangent
The stated point must lie on both the curve and the tangent line.
The tangent should touch the curve at that point and share the same gradient there.
Substitution provides a quick algebraic check.
Exponential-Decay Application
Exercise 10.2 uses an exponential model for a decreasing number of students.
The model has the form:
\(S=1000e^{-t}\)
The graph decreases quickly, remains non-negative, and approaches the horizontal axis.
Each increase of one time unit multiplies the previous value by:
\(e^{-1}\approx0.368\)
Demand and Supply Graphs
Demand and supply functions can be drawn on the same axes.
A decreasing demand line and an increasing supply line meet at the market equilibrium.
The equilibrium is found algebraically by setting:
\(P_d=P_s\)
Intersection of Two Graphs
The coordinates of an intersection satisfy both equations.
Set the two function expressions equal, solve for the independent variable, and substitute it into either equation to find the common value.
Salary Graph
A salary model with a fixed annual increase is linear.
For:
\(S(x)=45000+4500x\)
the constant term represents the starting salary.
The coefficient of \(x\) represents the annual increase.
Cost and Revenue Functions
A cost function may contain a fixed cost and a variable cost.
A revenue function depends on the number of items sold.
For example:
\(C(x)=1200+20x\)
and:
\(R(x)=50x\)
Break-Even Point
The break-even point occurs where cost equals revenue.
\(C(x)=R(x)\)
At this point, there is neither profit nor loss.
Graphically, it is the intersection of the cost and revenue graphs.
Profit and Loss
Profit is calculated by:
\(\text{Profit}=R(x)-C(x)\)
A positive result is profit, while a negative result is loss.
Before the break-even point, cost is greater than revenue.
After the break-even point, revenue is greater than cost.
Linear Profit Function
A profit function such as:
\(p(x)=10x-70\)
has constant gradient.
The gradient shows the additional profit from each extra unit.
Its horizontal intercept marks the point where profit first becomes zero.
Quadratic Cost Function
Exercise 10.2 includes a cost model containing a quadratic term:
\(C(x)=1500+10x+0.2x^2\)
Its graph becomes steeper as production increases because the rate of cost increase is not constant.
Printing Note in Exercise 10.2
The final cost question states a graphing interval ending at 150 shirts but asks for the cost of 200 shirts.
The solution identifies this inconsistency and extends the same formula to \(x=200\) to answer the stated question.
Solutions of Review Exercise 10 Unit 10 Class 9 Math Notes
Review Exercise 10 revises graph types, graph features, exponential and reciprocal functions, quadratic and cubic graphs, applications, intersections, and break-even analysis.
Identifying Types of Graphs
The multiple-choice section checks whether a function is linear, quadratic, cubic, exponential, or reciprocal.
The position of the variable and its highest power help identify the function type.
Vertical Lines
A vertical line has the equation:
\(x=a\)
It is parallel to the vertical axis and has undefined slope.
Exponential Growth and Decay
For:
\(y=a^x\)
the graph shows growth when \(a>1\).
A function such as:
\(y=3^{-x}=\left(\frac{1}{3}\right)^x\)
shows exponential decay.
Reciprocal-Graph Revision
The review includes:
\(y=\frac{2}{x},\qquad x\neq0\)
The graph has two branches in the first and third quadrants.
Its asymptotes are:
\(x=0,\qquad y=0\)
Limiting Exponential Model
The review uses a magazine-sales model:
\(S=200000\left(1-e^{-0.05t}\right)\)
The graph starts at zero, rises rapidly, and gradually levels off near 200,000.
The value 200,000 is the limiting level approached by the model.
Comparing Upward and Downward Parabolas
The review compares:
\(y=x^2-3\)
and:
\(y=15-x^2\)
The first opens upward and has a minimum point.
The second opens downward and has a maximum point.
Both are symmetric about the vertical axis.
Roots and Vertices of Simple Parabolas
For:
\(y=x^2-3\)
the vertex is \((0,-3)\), and the roots are obtained from:
\(x^2-3=0\)
For:
\(y=15-x^2\)
the vertex is \((0,15)\), and its roots are found from:
\(15-x^2=0\)
Cubic Graph in Factored Form
The review includes a cubic written as:
\(y=\frac{1}{2}(x+4)(x-1)(x-3)\)
The factored form gives three horizontal intercepts immediately.
A value table is then used to show the shape of the complete curve.
Intersection of Two Quadratic Graphs
Two quadratic functions can be graphed on the same axes.
Their intersection is found by setting the two expressions equal.
Substitution then gives the common vertical coordinate.
Television Cost and Revenue Application
The final review question compares a television company’s cost and revenue.
The break-even point is found by solving:
\(60000+250x=1200x\)
The graph shows loss before the intersection and profit after it.
Because the number of televisions must be a whole number, the change from loss to profit occurs between the two nearest whole-number quantities around the exact intersection.
Important Rules and Formulas of Unit 10 Class 9 Math
Straight-Line Form
\(y=mx+c\)
Axis of Symmetry of a Quadratic
\(x=-\frac{b}{2a}\)
Horizontal Intercept
\(f(x)=0\)
Vertical Intercept
\(f(0)\)
Gradient of a Tangent
\(m=f'(a)\)
Tangent Equation
\(y-f(a)=m(x-a)\)
Break-Even Condition
\(C(x)=R(x)\)
Profit Formula
\(\text{Profit}=R(x)-C(x)\)
Exponential Function
\(y=a^x\)
Basic Reciprocal Function
\(y=\frac{k}{x}\)
Square-Root Function
\(y=\sqrt{x}\)
Cube-Root Function
\(y=\sqrt[3]{x}\)
Common Mistakes in Unit 10 Class 9 Math Notes
Students may use unequal scales or irregular intervals on the coordinate axes.
Calculated points are sometimes plotted in the wrong quadrant.
Straight-line points should not be joined with a curved line.
Polynomial and exponential graphs should not be joined by separate straight segments.
Students may forget to calculate both horizontal and vertical intercepts.
The axis of symmetry is sometimes confused with a root of the quadratic.
The vertex must be calculated by substituting the axis-of-symmetry value into the function.
A reciprocal graph must not be joined through a value where the function is undefined.
Asymptotes should be shown as lines that the graph approaches but does not touch in the ordinary examples used here.
The excluded value in the denominator must be removed from the domain.
The horizontal-asymptote value may need to be excluded from the range.
Students sometimes confuse exponential notation with multiplication.
A tangent gradient is found at one specific point, not over a wide interval.
The tangent point should be checked in both the curve equation and tangent equation.
In application questions, the intersection must be interpreted according to the context.
Profit is revenue minus cost, not cost minus revenue.
Graph units, labels, and meaningful intervals should be written clearly.
Exam Preparation Tips for Unit 10 Class 9 Math Notes
Learn how to prepare a clear table of values.
Practise finding horizontal and vertical intercepts before plotting.
Memorize the straight-line form and axis-of-symmetry formula.
Learn the basic shapes of linear, quadratic, cubic, exponential, reciprocal, square-root, and cube-root graphs.
State domain, range, and asymptotes for exponential and reciprocal functions.
Use smooth curves for non-linear graphs.
Mark the vertex and roots of every parabola where possible.
Practise differentiating simple quadratic functions for tangent questions.
Use point-slope form to write tangent equations.
Learn how to find the intersection of two functions algebraically.
Understand the meanings of starting value, gradient, equilibrium, break-even, profit, and limiting value.
Attempt Review Exercise 10 independently before checking the solution PDF.
Use the save or download option in the PDF viewer to keep the solutions for offline revision.
Why Unit 10 Class 9 Math Solutions Are Important
Unit 10 Class 9 Math Solutions are important because graphs convert equations and numerical data into visual information.
Graphs make it easier to understand growth, decay, turning points, intercepts, rates of change, and the relationship between two quantities.
The chapter connects mathematics with economics, business, population models, salaries, costs, revenues, and other practical situations.
These graphing skills prepare students for coordinate geometry, calculus, statistics, physics, economics, and computer-based data analysis.
FAQs About Unit 10 Class 9 Math Solutions
What is the topic of Unit 10 Class 9 Math?
The topic of Unit 10 Class 9 Math is Graphs of Functions.
How many exercises are included in Unit 10?
Unit 10 includes Exercise 10.1, Exercise 10.2, and Review Exercise 10.
What is covered in Exercise 10.1?
Exercise 10.1 covers linear, quadratic, cubic, exponential, reciprocal, square-root, cube-root, and inverse-square graphs.
What is covered in Exercise 10.2?
Exercise 10.2 covers tangents, gradients, exponential models, demand and supply, salary graphs, cost and revenue, break-even points, profit, and quadratic cost.
What is covered in Review Exercise 10?
Review Exercise 10 revises graph types, exponential decay, reciprocal graphs, parabolas, cubic graphs, graph intersections, and break-even applications.
What is the form of a straight-line equation?
The standard form used in this unit is \(y=mx+c\), where \(m\) is the slope and \(c\) is the vertical intercept.
How is the axis of symmetry of a quadratic found?
For \(y=ax^2+bx+c\), the axis of symmetry is \(x=-\frac{b}{2a}\).
How is a horizontal intercept found?
Put the vertical coordinate equal to zero and solve the resulting equation.
How is a vertical intercept found?
Put the horizontal coordinate equal to zero and calculate the function value.
What is an asymptote?
An asymptote is a line that a graph approaches. Reciprocal functions may have both vertical and horizontal asymptotes.
How is the gradient of a tangent found?
For \(y=f(x)\), differentiate the function and substitute the required value into \(f'(x)\).
What is a break-even point?
The break-even point is the point where cost and revenue are equal, so there is neither profit nor loss.
Can students download the Unit 10 solutions PDFs?
Yes. Students can view the exercise-wise PDFs on this page and use the save or download option provided by the PDF viewer.
Are these Unit 10 Class 9 Math Solutions useful for exam preparation?
Yes. The solutions are useful for homework, revision, class tests, annual examinations, and board exam preparation.
Disclaimer
These Unit 10 Class 9 Math Solutions are provided for educational help. Students should use them to understand graphing methods, check their work, and study alongside the official textbook and their teacher’s instructions.
Where a printed interval or question appears inconsistent, the solution follows the stated formula and explains the issue.
Final Words
Unit 10 Class 9 Math Solutions help students understand Graphs of Functions in a clear and organized way.
Exercise 10.1 develops graph-sketching skills, Exercise 10.2 explains graphical applications and tangents, and Review Exercise 10 revises the complete chapter.
Prepare accurate value tables, label every axis, mark important features, and interpret each graph according to the question.