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Quadratic Functions Graph Sketching and Equation Finding
Published on July 11, 2026
Functions
Learn to sketch quadratic function graphs using factorization and find function equations from given graph properties like intercepts and vertex.
Question
Answer
To solve the math problems presented in the image, I will provide step-by-step solutions for each question.
1. Sketch the graph of the following functions using the method of factorization.
Since I cannot draw a physical sketch, I will provide the key features (x-intercepts, y-intercept, and vertex/direction) needed to sketch them.
(i)
- x-intercepts: Set
and . Points: . - y-intercept: Set
. Point: . - Direction: Since the leading coefficient (
) is positive, the parabola opens upwards. - Vertex: The x-coordinate is the midpoint of intercepts:
. Vertex is .
(ii)
- x-intercepts:
and . Points: . - y-intercept:
. Point: . - Direction: Leading coefficient (
) is negative, so it opens downwards. - Vertex:
. . Vertex: .
(iii)
- x-intercepts:
; . Points: . - y-intercept:
. Point: . - Direction: Opens upwards.
- Vertex:
. Vertex: .
(iv)
- Factorization:
. - x-intercepts:
and . Points: . - y-intercept:
. Point: . - Direction: Opens upwards.
(v)
- Factorization:
. - x-intercept:
(touches the axis). Point: . - y-intercept:
. Point: . - Direction: Opens upwards. Vertex is
.
(vi)
- x-intercept:
(touches the axis). Point: . - y-intercept:
. Point: . - Direction: Opens upwards. Vertex is
.
2. Find the functions whose graphs are shown in the following figures.
(i) Graph analysis:
- x-intercepts:
and . This means the equation is . - Point on graph:
. - Solve for
:
. - Equation:
.
(ii) Graph analysis:
- x-intercept:
. Since it touches the axis, it’s a double root: . - Point on graph:
. - Solve for
:
. - Equation:
.
3. Find an equation of the graph whose x-intercepts are
- Form:
. - Use point
:
. - Equation:
.
4. Find an equation of the graph which touches x-axis at
- Form (touches axis):
. - Use y-intercept
:
. - Equation:
.
5. Find an equation of the function whose graph cuts x-axis at
- Form:
. - Use vertex
:
. - Equation:
.