A reliable method
For a quadratic in vertex form a(x − h)² + k, identify the vertex (h,k) and the direction of opening. Use selected input-output pairs to check a transformation.
The common trap
The sign inside x − h can look opposite to the horizontal movement: a positive h shifts the graph right. Read the input that makes the inner expression zero.
Check and transfer
Intercepts and vertices solve different conditions. For a y-intercept, set x to zero; for an x-intercept, set the function value to zero.
Worked comparison: identify the structure
The vertex of y=2(x+3)²−5 is (−3,−5), because x=−3 makes the square zero. Substituting x=0 instead finds y=13, the vertical intercept. Vertex input, vertex output and intercept each solve a different condition.
Transfer and explain
Let h(x)=x² for x≥2 and h(x)=3x otherwise. At x=2, use the inclusive quadratic branch, giving 4. Applying the other branch gives 6 and violates the stated condition.
The function y = 2(x + 3)² − 5 has its vertex at (h,k). What is h?
- The inner expression equals zero at the vertex.
- Solve x plus the constant equal to zero; the positive sign inside indicates a left shift.
- The requested value is -3. Substitute or reverse the operation to check this result.
Compare the other answer choices
Assume vertical stretching leaves the vertex at input zero despite the horizontal shift.
Read the inner plus sign as a rightward rather than leftward shift.
Report the y-intercept instead of the vertex input.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
CLEP College Algebra exam overviewSiyavula Grade 10 foundation reference