209. Interpreting Model Graphs

Learning Intentions

• Identify intercepts, turning points and intervals of increase or decrease.
• Interpret Draw features in real-world contexts.
• Use graphs to estimate solutions to practical problems.

Pre-requisite Summary

• Know that an -intercept is where a graph crosses the -axis.
• Know that a -intercept is where a graph crosses the -axis.
• Know that a turning point is where a graph changes direction.
• Know that a graph is increasing when it rises from left to right and decreasing when it falls from left to right.
• Know that graph features can represent practical information, such as starting values, maximum heights or break-even points.
• Know that graph estimates are approximate and should be interpreted with units and context.

Worked Examples

Worked Example 1

For the graph of , identify:

a) the -intercepts

b) the -intercept

c) the turning point

Worked Example 2

For the graph of , identify:

a) the -intercepts

b) the -intercept

c) the turning point

Worked Example 3

The graph of a quadratic has turning point and opens upward.

a) State the interval where the graph is decreasing.

b) State the interval where the graph is increasing.

c) Explain how the turning point helps you decide.

Worked Example 4

The graph of a quadratic has turning point and opens downward.

a) State the interval where the graph is increasing.

b) State the interval where the graph is decreasing.

c) State whether the turning point is a maximum or minimum.

Worked Example 5

A ball’s height is modelled by a graph of , where is height in metres and is time in seconds.

Use the graph to interpret:

a) the -intercept

b) the turning point

c) the time when the ball returns to the ground

Worked Example 6

A company’s profit is modelled by a graph where is profit in dollars and is the number of items sold.

The graph has -intercepts at and , and a turning point at .

a) Interpret the -intercepts in context.

b) Interpret the turning point in context.

c) State the interval where profit is increasing.

Worked Example 7

Use the graph of to estimate the solutions to:

a)

b)

c)

Worked Example 8

A water fountain’s height is shown on a graph, where is height in metres and is time in seconds.

Use the graph to estimate:

a) the maximum height

b) when the water reaches metres

c) when the water returns to the ground

Problems

Problem 1

For the graph of , identify:

a) the -intercepts

b) the -intercept

c) the turning point

Problem 2

For the graph of , identify:

a) the -intercepts

b) the -intercept

c) the turning point

Problem 3

The graph of a quadratic has turning point and opens upward.

a) State the interval where the graph is decreasing.

b) State the interval where the graph is increasing.

c) Explain how the turning point helps you decide.

Problem 4

The graph of a quadratic has turning point and opens downward.

a) State the interval where the graph is increasing.

b) State the interval where the graph is decreasing.

c) State whether the turning point is a maximum or minimum.

Problem 5

A ball’s height is modelled by a graph of , where is height in metres and is time in seconds.

Use the graph to interpret:

a) the -intercept

b) the turning point

c) the time when the ball returns to the ground

Problem 6

A company’s profit is modelled by a graph where is profit in dollars and is the number of items sold.

The graph has -intercepts at and , and a turning point at .

a) Interpret the -intercepts in context.

b) Interpret the turning point in context.

c) State the interval where profit is increasing.

Problem 7

Use the graph of to estimate the solutions to:

a)

b)

c)

Problem 8

A water fountain’s height is shown on a graph, where is height in metres and is time in seconds.

Use the graph to estimate:

a) the maximum height

b) when the water reaches metres

c) when the water returns to the ground

Exercises

Understanding and Fluency

Exercise 1

For each graph feature, state what it means.

a) -intercept

b) -intercept

c) turning point

d) interval of increase

Exercise 2

For each quadratic graph, identify whether the turning point is a maximum or minimum.

a) A parabola opening upward with turning point

b) A parabola opening downward with turning point

c) A parabola opening upward with turning point

d) A parabola opening downward with turning point

Exercise 3

For each graph, state the interval where the graph is increasing and decreasing.

a) A parabola opening upward with turning point

b) A parabola opening upward with turning point

c) A parabola opening downward with turning point

d) A parabola opening downward with turning point

Exercise 4

Identify the -intercepts and -intercept from each description.

a) The graph crosses the -axis at and , and crosses the -axis at .

b) The graph crosses the -axis at and , and crosses the -axis at .

c) The graph touches the -axis at and crosses the -axis at .

Exercise 5

For each graph, interpret the -intercept in context.

a) A savings graph has on the vertical axis and weeks on the horizontal axis. The -intercept is .

b) A height graph has on the vertical axis and time on the horizontal axis. The -intercept is .

c) A cost graph has on the vertical axis and months on the horizontal axis. The -intercept is .

Exercise 6

For each context, interpret the turning point.

a) A ball’s height graph has turning point .

b) A profit graph has turning point .

c) A cost graph has turning point and opens upward.

Exercise 7

Use the graph features to answer each question.

A quadratic graph has -intercepts at and , and a maximum turning point at .

a) Where is the graph above the -axis?

b) Where is the graph increasing?

c) Where is the graph decreasing?

Exercise 8

Use the graph features to answer each question.

A quadratic graph has -intercepts at and , and a minimum turning point at .

a) Where is the graph below the -axis?

b) Where is the graph decreasing?

c) Where is the graph increasing?

Exercise 9

Use a graph of to estimate the solutions.

a)

b)

c)

Exercise 10

Use a graph of to estimate the solutions.

a)

b)

c)

Reasoning

Exercise 11

Explain why the -intercepts of a height-time graph can represent times when an object is on the ground.

Exercise 12

Explain why a turning point on a downward-opening parabola represents a maximum value.

Exercise 13

A student says that the -intercept of a ball’s height graph tells when the ball lands.

Explain the mistake.

Exercise 14

A student says that a graph with turning point is always increasing after .

Explain why this depends on whether the parabola opens upward or downward.

Exercise 15

A profit graph has -intercepts at and .

a) Explain what the intercepts mean in context.

b) Explain why the business is making a profit between these values if the graph is above the -axis there.

Exercise 16

Decide whether each statement is true or false. Justify your answer.

a) The -intercept occurs when .

b) A quadratic graph can have two turning points.

c) A graph can be used to estimate solutions by finding where it reaches a given height or value.

Problem-solving

Exercise 17

A ball is thrown from a platform. Its height is modelled by the graph of , where is height in metres and is time in seconds.

Use the graph to estimate:

a) the starting height

b) the maximum height

c) when the ball hits the ground

Exercise 18

A business models profit using a quadratic graph.

The graph has -intercepts at and , and a turning point at .

a) Interpret both -intercepts.

b) Interpret the turning point.

c) State the interval of sales where profit is increasing.

d) State the interval of sales where profit is decreasing.

Exercise 19

A water tank’s volume is represented by a graph where is volume in litres and is time in minutes.

The graph starts at , decreases to a minimum at , then increases again.

a) Interpret the -intercept.

b) Interpret the turning point.

c) Estimate when the tank has litres if the graph crosses at and .

Exercise 20

Create your own practical graph interpretation problem.

Your response must include:

• a real-world context
• at least two intercepts or one intercept and one turning point
• one interval of increase or decrease
• one graph-based estimate
• a sentence interpreting the estimate in context

Potential Misunderstandings

• Students may confuse -intercepts and -intercepts.
• Students may think an intercept is any point on the graph rather than a point where the graph crosses an axis.
• Students may forget that the -intercept occurs when the input value is .
• Students may think every turning point is a maximum.
• Students may describe increase and decrease by looking from right to left instead of left to right.
• Students may not connect intervals of increase or decrease to the -values on either side of the turning point.
• Students may read exact solutions from a graph when only estimates are possible.
• Students may give graph-based answers without units or context.
• Students may use parts of a graph outside the practical domain, such as negative time, negative sales or negative length.

Next; 210. Evaluating Applied Models