Grade 9 · Connect symmetry and quadratic values · 387 of 750
Connect symmetry and quadratic values · practice 42
For y = 0.5(x − (-3))² + (-4), find y at x = 2 and identify the vertex. Follow the calculation, then test the quantities in the animated example.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
For y = 0.5(x − (-3))² + (-4), find y at x = 2 and identify the vertex.
Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.
A starting point
Evaluate the whole difference before squaring; read the shifts from vertex form.
Work through the reasoning
Step 1
Translate the givens
At x = 2, the inner difference is 2 − (-3) = 5.
Step 2
Calculate with the model
Square it: (5)² = 25; then y = 0.5 × 25 + (-4) = 8.5.
Step 3
Check and interpret
At x = -3, the squared term is zero and y = -4, so the vertex is (-3, -4). Because 0.5 > 0 it is a minimum.
The answer
y(2) = 8.5; vertex = (-3, -4).
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
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The mathematical idea
Vertex form separates opening and curvature from horizontal and vertical shifts. Equal offsets on either side of the symmetry line give equal outputs. When the leading coefficient is zero, the rule is constant rather than quadratic, and it has no unique vertex. Starting quantities: Leading coefficient a = 0.5; Horizontal shift h = -3; Vertical shift k = -4.
y = a(x−h)²+k
03 · Reflect and transfer
Explain what changes and why.
Why do inputs equally spaced on either side of the vertex have the same output?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.