Mathematics
Grade 9
15 min
Identify linear quadratic and exponential functions from graphs
Identify linear quadratic and exponential functions from graphs
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1
Introduction & Learning Objectives
Learning Objectives
Distinguish between the graphical shapes of linear, quadratic, and exponential functions.
Identify a linear function by its straight-line graph and constant rate of change.
Identify a quadratic function by its symmetrical, U-shaped parabolic graph and vertex.
Identify an exponential function by its J-shaped or L-shaped curve that approaches a horizontal asymptote.
Describe the key visual characteristics of each function type.
Analyze a given graph and correctly classify the function it represents as linear, quadratic, or exponential, providing a reason for their choice.
Ever noticed how a thrown basketball, a savings account, and a straight road all follow different mathematical paths? 🏀💰🛣️ Let's learn how to spot them on a graph!
In this tutorial...
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Key Concepts & Vocabulary
TermDefinitionExample
Linear FunctionA function that creates a perfectly straight line when graphed. It has a constant rate of change, also known as slope.The cost of buying movie tickets at $15 each. The graph is a straight line that goes up by 15 for every ticket purchased.
Quadratic FunctionA function that creates a symmetrical, U-shaped curve called a parabola when graphed. The curve can open upwards or downwards.The path of a ball thrown into the air. It goes up, reaches a maximum height (the vertex), and comes back down, forming a U-shape.
Exponential FunctionA function where the variable is in the exponent. Its graph is a curve that increases or decreases very rapidly on one side, while approaching a horizontal line (an asymptote) on the other.The number of bacteria in a lab sample...
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Core Formulas
Linear Function Form
y = mx + b
The graph of an equation in this form is always a straight line. 'm' represents the constant slope, and 'b' is the y-intercept. A constant slope is the key feature of a linear graph.
Quadratic Function Form
y = ax^2 + bx + c (where a ≠ 0)
The graph of this equation is a parabola. If 'a' is positive, the U-shape opens upwards. If 'a' is negative, it opens downwards. The presence of the x^2 term creates the curve.
Exponential Function Form
y = ab^x (where a ≠ 0, b > 0, b ≠ 1)
The graph is a curve showing growth (if b > 1) or decay (if 0 < b < 1). The variable 'x' in the exponent causes the rapid change and the horizontal asymptote.
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Challenging
A function's graph exists only in Quadrants I and II and approaches the x-axis as x approaches negative infinity. Which function type is it impossible for this graph to be?
A.Exponential with base > 1
B.Quadratic opening upwards with vertex in Q1 or Q2
C.Exponential with base between 0 and 1
D.linear function with a non-zero slope
Challenging
The rate of change between points on a graph is not constant, but the graph does have a clear axis of symmetry. What can you conclude about the function?
A.The function must be quadratic.
B.The function must be exponential.
C.The function must be linear.
D.The function could be either quadratic or exponential.
Challenging
Graph A is a curve through (-2, 9), (0, 5), and (2, 9). Graph B is a curve through (0, 5), (1, 10), and (2, 20). Which statement is correct?
A.Both are quadratic.
B.Both are exponential.
C.Graph A is quadratic, and Graph B is exponential.
D.Graph A is exponential, and Graph B is quadratic.
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