Questions: Graphing Quadratic Functions: Vertex and Intercepts
5 questions to test your understanding
Score: 0 / 5
Question 1 Multiple Choice
What is the vertex of the parabola f(x) = (x + 5)² − 3?
A(5, −3)
B(−5, −3)
C(5, 3)
D(−5, 3)
In vertex form f(x) = a(x − h)² + k, the vertex is at (h, k). To read h correctly, rewrite (x + 5)² as (x − (−5))², so h = −5 and k = −3, giving vertex (−5, −3). The most common error is reading the vertex as (5, −3), taking the sign that appears in the expression rather than the actual value of h. The vertex is where the squared term equals zero, which happens when x = h = −5.
Question 2 Multiple Choice
Before solving, you calculate the discriminant of a quadratic and get −4. What does this tell you about the graph?
AThe parabola opens downward
BThe vertex is below the x-axis
CThe parabola has no x-intercepts and lies entirely above or below the x-axis
DThe parabola touches the x-axis at exactly one point
A negative discriminant (b² − 4ac < 0) means the quadratic equation has no real solutions — the roots are complex. Graphically, this means the parabola never crosses the x-axis. Whether it lies entirely above or below depends on the sign of a: if a > 0 (opens upward) and the vertex is above the x-axis, it stays above; if a < 0 (opens downward) and the vertex is below, it stays below. A discriminant of zero means exactly one x-intercept (tangent touch); positive means two x-intercepts.
Question 3 True / False
The axis of symmetry of a parabola always passes through the vertex.
TTrue
FFalse
Answer: True
By definition, the axis of symmetry is the vertical line x = h, where h is the x-coordinate of the vertex. Every parabola is symmetric about this line — if you fold the parabola along it, both halves match perfectly. This relationship also explains why the axis of symmetry is the midpoint of the two x-intercepts when they exist: the x-intercepts are equidistant from the vertex on either side.
Question 4 True / False
A parabola with a positive leading coefficient (a > 0) generally has two x-intercepts.
TTrue
FFalse
Answer: False
The number of x-intercepts depends on the discriminant, not the sign of a. A parabola with a > 0 opens upward, but if its vertex is above the x-axis (k > 0), it never crosses the x-axis and has no real x-intercepts. For example, f(x) = x² + 1 has vertex (0, 1), opens upward, and has no x-intercepts. The sign of a determines direction of opening, not the number of intercepts.
Question 5 Short Answer
Without solving the quadratic, explain how the discriminant lets you predict the shape and position of the graph relative to the x-axis.
Think about your answer, then reveal below.
Model answer: The discriminant b² − 4ac counts the real solutions to f(x) = 0, which are the x-intercepts. If positive: two real roots → parabola crosses the x-axis at two points. If zero: one repeated root → parabola touches the x-axis at exactly one point (the vertex is on the x-axis). If negative: no real roots → parabola doesn't cross the x-axis at all, lying entirely above (a > 0) or below (a < 0) it. Combined with the sign of a (direction) and the vertex coordinates, this gives a complete picture of the graph before any solving.
The discriminant is powerful because it answers the qualitative question ('how does this parabola relate to the x-axis?') instantly, without the work of the quadratic formula. In applied contexts — like determining whether a projectile reaches a certain height — this qualitative answer is often all you need.