Question

practice
what is the y-intercept of each function?
see example 1

1. $f(x) = 2x^2 - 4x - 6$ 16. $f(x) = 0.3x^2 + 0.6x - 0.7$
2. $f(x) = -2x^2 - 8x - 7$ 18. $f(x) = 3x^2 + 6x + 5$
3. $f(x) = -x^2 - 2x + 3$ 20. $f(x) = -0.5x^2 + x + 2$

find the y-intercept, the axis of symmetry, and the vertex of the graph of each function. see example 2

1. $f(x) = 2x^2 + 8x + 2$ 22. $f(x) = -2x^2 + 4x - 3$
2. $f(x) = 0.4x^2 + 1.6x$ 24. $f(x) = -x^2 - 2x - 5$
3. $f(x) = 5x^2 + 5x + 12$ 26. $f(x) = 4x^2 + 12x + 5$
4. $f(x) = x^2 - 6x + 12$ 28. $f(x) = -2x^2 + 16x + 40$

Explanation:

Step1: Recall y-intercept definition

The y - intercept of a function \( f(x) \) is the value of \( f(x) \) when \( x = 0 \), i.e., \( f(0) \).

Step2: Solve for question 15 (\( f(x)=2x^{2}-4x - 6 \))

Substitute \( x = 0 \) into the function:
\( f(0)=2(0)^{2}-4(0)-6=-6 \)

Step3: Solve for question 16 (\( f(x)=0.3x^{2}+0.6x - 0.7 \))

Substitute \( x = 0 \):
\( f(0)=0.3(0)^{2}+0.6(0)-0.7=-0.7 \)

Step4: Solve for question 17 (\( f(x)=-2x^{2}-8x - 7 \))

Substitute \( x = 0 \):
\( f(0)=-2(0)^{2}-8(0)-7=-7 \)

Step5: Solve for question 18 (\( f(x)=3x^{2}+6x + 5 \))

Substitute \( x = 0 \):
\( f(0)=3(0)^{2}+6(0)+5 = 5 \)

Step6: Solve for question 19 (\( f(x)=-x^{2}-2x + 3 \))

Substitute \( x = 0 \):
\( f(0)=-(0)^{2}-2(0)+3=3 \)

Step7: Solve for question 20 (\( f(x)=-0.5x^{2}+x + 2 \))

Substitute \( x = 0 \):
\( f(0)=-0.5(0)^{2}+0 + 2=2 \)

Step8: Solve for question 21 (\( f(x)=2x^{2}+8x + 2 \))

Find y - intercept:

Substitute \( x = 0 \): \( f(0)=2(0)^{2}+8(0)+2 = 2 \)

Find axis of symmetry:

For a quadratic function \( f(x)=ax^{2}+bx + c \), the axis of symmetry is \( x=-\frac{b}{2a} \). Here \( a = 2 \), \( b = 8 \), so \( x=-\frac{8}{2\times2}=-2 \)

Find vertex:

The x - coordinate of the vertex is the axis of symmetry \( x=-2 \). Substitute \( x=-2 \) into \( f(x) \):
\( f(-2)=2(-2)^{2}+8(-2)+2=2\times4-16 + 2=8-16 + 2=-6 \). So vertex is \( (-2,-6) \)

Step9: Solve for question 22 (\( f(x)=-2x^{2}+4x - 3 \))

Find y - intercept:

Substitute \( x = 0 \): \( f(0)=-2(0)^{2}+4(0)-3=-3 \)

Find axis of symmetry:

\( a=-2 \), \( b = 4 \), axis of symmetry \( x=-\frac{4}{2\times(-2)}=1 \)

Find vertex:

Substitute \( x = 1 \) into \( f(x) \): \( f(1)=-2(1)^{2}+4(1)-3=-2 + 4-3=-1 \). Vertex is \( (1,-1) \)

Step10: Solve for question 23 (\( f(x)=0.4x^{2}+1.6x \))

Find y - intercept:

Substitute \( x = 0 \): \( f(0)=0.4(0)^{2}+1.6(0)=0 \)

Find axis of symmetry:

\( a = 0.4 \), \( b = 1.6 \), axis of symmetry \( x=-\frac{1.6}{2\times0.4}=-\frac{1.6}{0.8}=-2 \)

Find vertex:

Substitute \( x=-2 \) into \( f(x) \): \( f(-2)=0.4(-2)^{2}+1.6(-2)=0.4\times4-3.2 = 1.6-3.2=-1.6 \). Vertex is \( (-2,-1.6) \)

Step11: Solve for question 24 (\( f(x)=-x^{2}-2x - 5 \))

Find y - intercept:

Substitute \( x = 0 \): \( f(0)=-(0)^{2}-2(0)-5=-5 \)

Find axis of symmetry:

\( a=-1 \), \( b=-2 \), axis of symmetry \( x=-\frac{-2}{2\times(-1)}=\frac{2}{-2}=-1 \)

Find vertex:

Substitute \( x=-1 \) into \( f(x) \): \( f(-1)=-(-1)^{2}-2(-1)-5=-1 + 2-5=-4 \). Vertex is \( (-1,-4) \)

Step12: Solve for question 25 (\( f(x)=5x^{2}+5x + 12 \))

Find y - intercept:

Substitute \( x = 0 \): \( f(0)=5(0)^{2}+5(0)+12 = 12 \)

Find axis of symmetry:

\( a = 5 \), \( b = 5 \), axis of symmetry \( x=-\frac{5}{2\times5}=-\frac{1}{2}=-0.5 \)

Find vertex:

Substitute \( x=-0.5 \) into \( f(x) \): \( f(-0.5)=5(-0.5)^{2}+5(-0.5)+12=5\times0.25-2.5 + 12=1.25-2.5 + 12=10.75 \). Vertex is \( (-0.5,10.75) \)

Step13: Solve for question 26 (\( f(x)=4x^{2}+12x + 5 \))

Find y - intercept:

Substitute \( x = 0 \): \( f(0)=4(0)^{2}+12(0)+5 = 5 \)

Find axis of symmetry:

\( a = 4 \), \( b = 12 \), axis of symmetry \( x=-\frac{12}{2\times4}=-\frac{12}{8}=-\frac{3}{2}=-1.5 \)

Find vertex:

Substitute \( x=-1.5 \) into \( f(x) \): \( f(-1.5)=4(-1.5)^{2}+12(-1.5)+5=4\times2.25-18 + 5=9-18 + 5=-4 \). Vertex is \( (-1.5,-4) \)

Step14: Solve for question 27 (\( f(x)=x^{2}-6x + 12 \))

Find y - intercept:

Substitute \( x = 0 \): \( f(0)=0^{2}-6(0)+12 = 12 \)

Find axis of symmetry:

\( a = 1 \), \( b=-6 \), axis of symmetry \( x=-\frac{-6}{2\times1}=3 \)

Find vertex:

Substitute \( x = 3 \) into \( f(x) \): \( f(3)=3^{2}-6\times3 + 12=9-18 + 12=3 \). Vertex is \( (3,3) \)

Step15: Solve for question 28 (\( f(x)=-2x^{2}+16x + 40 \))

Find y - intercept:

Substitute \( x = 0 \): \( f(0)=-2(0)^{2}+16(0)+40 = 40 \)

Find axis of symmetry:

\( a=-2 \), \( b = 16 \), axis of symmetry \( x=-\frac{16}{2\times(-2)}=4 \)

Find vertex:

Substitute \( x = 4 \) into \( f(x) \): \( f(4)=-2(4)^{2}+16(4)+40=-32 + 64+40=72 \). Vertex is \( (4,72) \)

Answer:

1. \( -6 \)
2. \( -0.7 \)
3. \( -7 \)
4. \( 5 \)
5. \( 3 \)
6. \( 2 \)
7. y - intercept: \( 2 \), axis of symmetry: \( x=-2 \), vertex: \( (-2,-6) \)
8. y - intercept: \( -3 \), axis of symmetry: \( x = 1 \), vertex: \( (1,-1) \)
9. y - intercept: \( 0 \), axis of symmetry: \( x=-2 \), vertex: \( (-2,-1.6) \)
10. y - intercept: \( -5 \), axis of symmetry: \( x=-1 \), vertex: \( (-1,-4) \)
11. y - intercept: \( 12 \), axis of symmetry: \( x=-0.5 \), vertex: \( (-0.5,10.75) \)
12. y - intercept: \( 5 \), axis of symmetry: \( x=-1.5 \), vertex: \( (-1.5,-4) \)
13. y - intercept: \( 12 \), axis of symmetry: \( x = 3 \), vertex: \( (3,3) \)
14. y - intercept: \( 40 \), axis of symmetry: \( x = 4 \), vertex: \( (4,72) \)