Exponential Functions Primer

Overview

In this lesson (Lesson 5 of 9), students build on earlier exposure to exponential growth/decay by identifying key graph features and solving basic exponential equations using algebra and, when appropriate, technology. Students will move from qualitative reasoning (intercepts/asymptotes/transformations) to quantitative problem solving (equations).

Learning intentions

Students will:

• recognise and describe qualitative features of exponential graphs, including intercepts and asymptotes
• interpret growth and decay from the base and shape of exponential functions
• solve equations involving exponential expressions, with and without technology
• check solutions for reasonableness using graph/trace reasoning

Success criteria

Students can:

• state the y-intercept and the horizontal asymptote for simple exponential forms
• explain how parameters affect the graph in words (and match to a sketch)
• solve an exponential equation and verify the solution satisfies the original equation
• use technology to confirm solutions and make sense of whether there are 0, 1, or 2 solutions

Curriculum links

• Exponential functions — recognise and determine qualitative features of the graph of (y=r^x) (where (r>0)), including asymptote and intercept
• Exponential functions — solve equations involving exponential functions, with and without technology
• Exponential functions — sketch graphs of exponential functions, with and without technology
• Exponential functions — model and solve problems involving exponential functions, with and without technology

Lesson structure (60 minutes)

1. 0–7 min · Starter (growth/decay quick-sort). Teacher shows 3 equations (e.g., (y=2^x), (y=(1/2)^x), (y=3^{x-1}+2)) and asks students to predict which graphs show growth/decay before any sketching. Students sort them into “growth/decay/mixed/shifted” and justify briefly in one sentence each.

2. 7–18 min · Direct teach (key features from form). Teacher models how to read intercepts and horizontal asymptotes from the structure of exponential functions, starting with (y=r^x) and then (y=r(x-h)+k) style form. Students complete a short teacher-guided table for each function: “y-intercept”, “asymptote”, “is it growth or decay?” and “how do shifts change the graph?”.

3. 18–28 min · Guided practice (sketch from features). Teacher provides two functions: one in the simple form (y=r^x), and one with a vertical shift, and demonstrates a quick sketch using asymptote and intercept first, then shape. Students sketch both graphs (no technology yet) and label the asymptote and intercept; they must write one line explaining why.

4. 28–40 min · Practice solving (same base equation). Teacher writes an equation such as (2^{x}=16) and then a slightly harder one such as (3^{x}=81), showing steps using logarithms only if needed, otherwise using known powers/rewriting. Students solve two equations independently, then one student shares a method while the class compares efficiency.

5. 40–50 min · Practice solving (equations with technology). Teacher introduces a technology check: students can solve numerically (or use graph intersection) and then verify by substitution. Students solve an equation like (2^{x}+2=10) or (3^{x}=2\cdot 9^{x-1}) (chosen so students can verify), using technology to confirm solutions and write the exact/numeric answer.

6. 50–58 min · Consolidation (reasonableness check). Teacher asks: “How can we tell if we have no, one, or two solutions?” using monotonicity of exponentials for (r>0) and the effect of shifts/scales. Students answer in pairs: for one given equation, state expected number of solutions and support it with graph/trace reasoning (no full solving required).

7. 58–60 min · Exit ticket (quick diagnostic). Teacher collects one final question: determine asymptote and intercept for (y=5^{x}-3), and solve a short equation such as (5^{x}=25). Students submit final answers with brief working/verification.

Resources

• Whiteboard/markers or interactive display
• Graph paper or pre-printed sketch grids
• Calculator or CAS/graphing technology (teacher-approved)
• Function cards for growth/decay sorting
• Short feature-reading table (intercept/asymptote/growth/decay)
• Exit ticket slips

Assessment

• Formative check during the starter sort: listen for correct growth vs decay reasoning
• Monitor guided sketching: students must label asymptote/intercept and justify shape
• Solution verification check: students show substitution or technology confirmation for at least one equation
• Exit ticket: intercept/asymptote identification plus one solved exponential equation

Differentiation

• Support: provide sentence starters such as “The horizontal asymptote is … because as (x\to -\infty) …” and “To solve, rewrite so the bases match …”
• Support: give students a worked example scaffold for steps: “rewrite → isolate exponent → solve → verify”
• Extension: ask students to solve an equation that requires noticing equivalence (e.g., expressing one side as a power of the other base) and to discuss why technology may show an extra apparent root if graphing window is poor
• EAL/SEN considerations: allow students to explain reasoning orally, and focus marking on method logic (asymptote/intercept + correct verification) rather than perfect algebra presentation