When the quadratic function is graphed in the -plane, where , its vertex is . One of the -intercepts of this graph is . What is the other -intercept of the graph?
A quadratic graph is a parabola, and its vertex gives the vertical axis of symmetry. If you know one -intercept, the other lies the same distance across that axis because both have height . Find the horizontal gap and go that far in the opposite direction. Reflecting across the -axis instead uses the wrong mirror line.
Hints
- Hint 1
A parabola has an axis of symmetry, a vertical mirror line through its vertex. Both -intercepts have height , so they sit the same horizontal distance from that line. Which coordinate of the vertex locates it?
- Hint 2
Write the vertex’s -coordinate and the known intercept’s -coordinate in fourths to find the gap between them. The known intercept is to the right of the axis. Which direction should you go to find its mirror?
Step-by-step
Reflect the intercept across the vertex
Step 1Find the mirror line
No Desmos needed. Symmetry gives the other intercept without finding the function. The vertex is the parabola’s turning point. Its -coordinate gives the axis of symmetry, the vertical mirror line . Both -intercepts sit equally far from this line because both have height . The vertex’s is a height, not a horizontal distance.
- Step 2
Measure the gap to the known intercept
Write the axis’s -coordinate in fourths to match the known intercept:
Subtract to find how far is to its right:
- Step 3
Go the same distance to the left
Go unit left of the axis, not right again:
An -intercept has -coordinate , so the other -intercept is . Choice D.