Exercises on Zero, Constant, Linear and Quadratic Functions
1. Definition of Linear Function. What conditions on 𝑚 and 𝑏 will ensure that 𝑓(𝑥) = 𝑚𝑥 + 𝑏 is a linear
function?
2. Characteristics of Zero, Constant and Linear Functions. Complete the table by giving the characteristics
of the given function.
Function
Slope
Domain
Range
𝒚-int/s
𝒙-int/s
Zeros Behavior
𝑭(𝒙) = 𝟎
𝑮(𝒙) = −√𝟕
𝒉(𝒙) = 𝟑𝒙 + 𝟐
𝒈(𝒙) = −𝒙 − 𝟒
𝒇(𝒙) = 𝟐𝒙 − 𝟓 on the set
{𝒙 ∈ 𝑹| − 𝟓 ≤ 𝒙 < 2}
3. Graphs of Linear Functions. Sketch the graph of the following linear functions. Label the intercepts.
𝒈(𝒙) = −𝒙 − 𝟒
𝒉(𝒙) = 𝟑𝒙 + 𝟐
𝒇(𝒙) = 𝟐𝒙 − 𝟓 on the set {𝒙 ∈ 𝑹| − 𝟓 ≤ 𝒙 < 2}
𝒇(𝒙) = −𝒙 + 𝟑 on the set {𝒙 ∈ 𝑹|𝒙 < 3 𝑜𝑟 𝑥 ≥ 5}
1|Page
4. More on Characteristics of the Zero, Constant and Linear Functions. Answer the following problems.
1. Determine the equation of the linear function ℎ 2. Find the value of 𝑎 if 𝑓(𝑥) = (𝒂 + 2)𝑥 + 3 and
whose 𝑦-intercept is the same as the 𝑦- intercept
𝑔(𝑥) = −2𝑥 + 7 are linear functions whose
of the constant function 𝑔(𝑥) = 5 and whose
graphs are perpendicular to each other.
graph is parallel to the graph of
𝑓(𝑥) = 2𝑥 − 5(𝑥 − 3).
3. Determine the equation of the linear function
whose zero is −5 and whose 𝑦-intercept is 3.
4. Determine the intersection of the graph of
𝑓(𝑥) = −5𝑥 + 12 and the constant function
𝑔(𝑥) = −3.
5. Which function will have a larger slope: an
increasing linear function or a decreasing linear
function?
6. If G is a constant function such that 𝐺(−3) = 2,
what is 𝐺(−1000)?
7. If 𝑓(𝑥) = 2𝑥 − 3 and 𝑔(𝑥) = 7𝑥 + 2, what is
the solution set of 𝑓(𝑥) > 𝑔(𝑥)
8. What is the solution set of ℎ(𝑥) ≥ −2, given
that ℎ(𝑥) = 3𝑥 − 1?
2|Page
5. Definition of Quadratic Function. What conditions on 𝑎, 𝑏, 𝑐 will ensure that 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 is a
quadratic function?
6. Vertex Form. Write the following in vertex form and give the vertex:
Quadratic
Vertex Form
Function
𝒈(𝒙) = 𝒙𝟐
Vertex
𝒎(𝒙) = 𝒙𝟐 − 𝟔𝒙 + 𝟏𝟏
𝒂(𝒙) = −𝒙𝟐 + 𝟒𝒙 + 𝟏
𝒕(𝒙) = −𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟑
𝒉(𝒙) = −𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟓
7. Concavity/Vertex/Domain/Range. Complete the table below and sketch the graph on a graphing paper:
(Fill up the upper 4x6 table vertically the remaining shaded cells horizontally. To practice your use of
vertex formula, except for m(x), use the vertex formula in getting the vertex of 𝑔, 𝑎, 𝑡 & ℎ, then verify
using the result of from the previous table.)
Quadratic
Concavity
Vertex
Domain
Range
𝒙-int/s
𝒚-int/s
Function
(upward or
downward)
𝒈(𝒙) = 𝒙𝟐
𝒎(𝒙) = (𝒙 − 𝟑)𝟐 + 𝟐
𝒂(𝒙) = −𝒙𝟐 + 𝟒𝒙 + 𝟏
𝒕(𝒙) = −𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟑
𝒉(𝒙) = −𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟓
8. Axis of Symmetry/Behavior. Complete the table below: (Fill up the upper 4x3 table vertically then the
remaining shaded cells horizontally.)
Quadratic
Axis of
Interval where graph is Interval where graph is
Function
Symmetry
increasing
decreasing
𝟐
𝒈(𝒙) = 𝒙
𝒎(𝒙) = (𝒙 − 𝟑)𝟐 + 𝟐
𝒂(𝒙) = −𝒙𝟐 + 𝟒𝒙 + 𝟏
𝒕(𝒙) = −𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟑
𝒉(𝒙) = −𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟓
3|Page
9. Discriminant and Zeros. Complete the table below: (Fill up the upper 4x3 table vertically then the
remaining shaded cells horizontally.)
Quadratic
Function
Discriminant
Nature of Zeros
Zeros
𝒈(𝒙) = 𝒙𝟐
𝒎(𝒙) = (𝒙 − 𝟑)𝟐 + 𝟐
𝒂(𝒙) = −𝒙𝟐 + 𝟒𝒙 + 𝟏
𝒕(𝒙) = −𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟑
𝒉(𝒙) = −𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟓
10. More on zeros. Find the zeros of the following. Use the quadratic formula for e and f:
a. 𝒇(𝒙) = 𝒙𝟐 + 𝟖𝟏
d. 𝒌(𝒙) = 𝟐(𝒙 − 𝟑)𝟐 + 𝟓
b. 𝒈(𝒙) = (𝒙 − 𝟔)𝟐 + 𝟐𝟎
a. 𝒉(𝒙) = −𝟐𝒙𝟐 + 𝒙 − 𝟑
c. 𝒉(𝒙) = −(𝒙 + 𝟐)𝟐 − 𝟐𝟒
b. 𝒎(𝒙) = 𝟑 𝒙𝟐 + 𝟐𝒙 + 𝟓
𝟏
4|Page
11. Graphs of QF. Sketch the graph of the following QF. Label the intercepts and the vertex. Draw also the
axis of symmetry.
𝒈(𝒙) = 𝒙𝟐
𝒎(𝒙) = (𝒙 − 𝟑)𝟐 + 𝟐
𝒂(𝒙) = −𝒙𝟐 + 𝟒𝒙 + 𝟏
𝒕(𝒙) = −𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟑
12. More on the Characteristics of QF. Find the quadratic function with the following characteristics:
 𝑥-intercepts of the function 𝐻 are 2, 3
 Zeros of the function 𝐺 are 3𝑖 and – 3𝑖
 𝑦-intercept is -3
 𝑦-intercept is 5



the axis of symmetry of function 𝐹 is 𝑥 = −2
there is only one 𝑥-intercept
(3, 4) is a point on the graph
 𝑦 −intercept of the function 𝑃 is 4
 𝑎 is 2
 (8,10) and (– 18, 10) are two points on its
graph
5|Page